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"location": "Some Arithmetical Questions", "latex": "in arithmetic an integral number is denoted by a succession of digits, where each digit represents the product of that digit and a power of ten, and the number is equal to the sum of these products.", "markdown": "in arithmetic an integral number is denoted by a succession of digits, where each digit represents the product of that digit and a power of ten, and the number is equal to the sum of these products.", "why": "It states the place-value idea behind all the scale-of-notation tricks, which a learner needs before following them.", "use": [ "lesson" ], "concepts": [ "concept/denary-scale-of-notation", "concept/digit" ] }, { "id": "ball-mathematical-recreations-1905/x-dab4fc1dc5", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 26", "location": "Some Arithmetical Questions", "latex": "Then the sum obtained as the result of this last operation will be $1089$.", "markdown": "Then the sum obtained as the result of this last operation will be $1089$.", "why": "It states the surprising fixed result of the reversed-digits trick, which makes a good hook for a learner to test and then explain.", "use": [ "website", "lesson" ], "concepts": [ "concept/digit", "concept/number" ] }, { "id": "ball-mathematical-recreations-1905/x-bfe63e6964", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 24", "location": "Some Arithmetical Questions", "latex": "Hence, if $N$ is divided by $a'$, the remainder is $a$.", "markdown": "Hence, if $N$ is divided by $a'$, the remainder is $a$.", "why": "It shows the key step by which a number is recovered from its remainders on division, which a learner can follow with small divisors.", "use": [ "lesson" ], "concepts": [ "concept/remainder" ] }, { "id": "ball-mathematical-recreations-1905/x-39bdb8d7de", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 30", "location": "Some Arithmetical Questions", "latex": "The result is the man's age in 1906.", "markdown": "The result is the man’s age in 1906.", "why": "It names the purpose of the age-from-birth-year trick, so a reader can see what the arithmetic is for.", "use": [ "website" ], "concepts": [ "concept/number" ] }, { "id": "ball-mathematical-recreations-1905/x-c62f2686eb", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 32", "location": "Some Arithmetical Questions", "latex": "The reason of the rule is obvious, for he arrives finally at the $(n + 12-m)$th hour from which he started.", "markdown": "The reason of the rule is obvious, for he arrives finally at the $(n + 12-m)$th hour from which he started.", "why": "It shows a learner why adding twelve hours changes nothing on a clock, which is the key idea of modular arithmetic.", "use": [ "lesson" ], "concepts": [ "concept/modular-arithmetic" ] }, { "id": "ball-mathematical-recreations-1905/x-952424a8ae", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 35", "location": "Some Arithmetical Questions", "latex": "Problems like this can be worked out only by trial: there are several solutions, of which one is as follows.", "markdown": "Problems like this can be worked out only by trial: there are several solutions, of which one is as follows.", "why": "It tells the learner plainly that some puzzles have no shortcut and must be solved by systematic testing.", "use": [ "lesson" ], "concepts": [ "method/trial-and-error" ] }, { "id": "ball-mathematical-recreations-1905/x-aef13e9452", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 36", "location": "Some Arithmetical Questions", "latex": "Obviously, if $A$ calls $43$, then whatever $B$ adds to that, $A$ can win next time.", "markdown": "Obviously, if $A$ calls $43$, then whatever $B$ adds to that, $A$ can win next time.", "why": "It gives a concrete winning position that a learner can check by hand, which introduces the idea of key numbers.", "use": [ "lesson", "website" ], "concepts": [ "concept/arithmetical-progression", "concept/winning-key-numbers" ] }, { "id": "ball-mathematical-recreations-1905/x-752e770190", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 39", "location": "Some Arithmetical Questions", "latex": "In other words, if the number of counters taken is expressed in the scale of notation whose radix is $n$, then the $(h + 1)$th digit from the right will give the number on the domino selected by $P_h$.", "markdown": "In other words, if the number of counters taken is expressed in the scale of notation whose radix is $n$, then the $(h + 1)$th digit from the right will give the number on the domino selected by $P_h$.", "why": "It explains how a positional scale can carry hidden information, which makes the digit idea concrete.", "use": [ "lesson" ], "concepts": [ "concept/denary-scale-of-notation", "concept/digit" ] }, { "id": "ball-mathematical-recreations-1905/x-4f9179e7d4", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 43", "location": "Some Arithmetical Questions", "latex": "The error in each of the foregoing examples is obvious, but the fallacies in the next examples are concealed somewhat better.", "markdown": "The error in each of the foregoing examples is obvious, but the fallacies in the next examples are concealed somewhat better.", "why": "It warns learners that a proof can look valid at every line, so each step must be checked.", "use": [ "lesson" ], "concepts": [ "concept/arithmetical-fallacy" ] }, { "id": "ball-mathematical-recreations-1905/x-f037a63be4", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 44", "location": "Some Arithmetical Questions", "latex": "Now if we put $a = d =1$ and $b = c = -1$ we have four numbers which satisfy the relation $ad = bc$ and such that $a>b$; hence, by the proposition, $c > d$, that is, $-1 > 1$, which is absurd.", "markdown": "Now if we put $a = d =1$ and $b = c = -1$ we have four numbers which satisfy the relation $ad = bc$ and such that $a>b$; hence, by the proposition, $c > d$, that is, $-1 > 1$, which is absurd.", "why": "It shows how a sound rule applied outside its conditions can yield an absurd result, a useful lesson in checking hypotheses.", "use": [ "history", "lesson" ], "concepts": [ "concept/arithmetical-fallacy", "person/jean-le-rond-d-alembert" ] }, { "id": "ball-mathematical-recreations-1905/x-ee523a62af", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 45", "location": "Some Arithmetical Questions", "latex": "To the above examples I may add the following questions, which I have often propounded in past years: though not fallacies, they may serve to illustrate the fact that the answer to an arithmetical question is frequently different to what a hasty reader might suppose.", "markdown": "To the above examples I may add the following questions, which I have often propounded in past years: though not fallacies, they may serve to illustrate the fact that the answer to an arithmetical question is frequently different to what a hasty reader might suppose.", "why": "It tells the learner up front that a confident first answer to an arithmetic question is often wrong, which is the habit the chapter wants to build.", "use": [ "lesson", "website" ], "concepts": [ "concept/arithmetic" ] }, { "id": "ball-mathematical-recreations-1905/x-a4d8ebac7d", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 45", "location": "Some Arithmetical Questions", "latex": "The answer is the latter; for in the first year the first clerk receives \\pounds100, but the second clerk receives \\pounds50 and \\pounds55 as his two half-yearly payments and thus receives in all \\pounds105.", "markdown": "The answer is the latter; for in the first year the first clerk receives 100, but the second clerk receives 50 and 55 as his two half-yearly payments and thus receives in all 105.", "why": "Working two salary schemes side by side shows a learner how a small change in payment timing can beat a larger headline rise.", "use": [ "lesson" ], "concepts": [ "concept/arithmetic" ] }, { "id": "ball-mathematical-recreations-1905/x-efa2315d9a", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 53", "location": "Some Arithmetical Questions", "latex": "It may be shown that $2^m +1$ is composite if $m$ is not a power of $2$, but of course it does not follow that $2^m + 1$ is a prime if $m$ is a power of $2$.", "markdown": "It may be shown that $2^m +1$ is composite if $m$ is not a power of $2$, but of course it does not follow that $2^m + 1$ is a prime if $m$ is a power of $2$.", "why": "It warns that a pattern holding for several cases does not guarantee it holds in general, a common mistake for beginners.", "use": [ "lesson" ], "concepts": [ "concept/fermat-number", "concept/prime-number" ] }, { "id": "ball-mathematical-recreations-1905/x-e4d4ea7064", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 54", "location": "Some Arithmetical Questions", "latex": "This proposition has acquired extraordinary celebrity from the fact that no general demonstration of it has been given, but there is no reason to doubt that it is true.", "markdown": "This proposition has acquired extraordinary celebrity from the fact that no general demonstration of it has been given, but there is no reason to doubt that it is true.", "why": "It gives a clear account of why Fermat's claim was famous: it was widely believed yet unproven for centuries.", "use": [ "history" ], "concepts": [ "person/pierre-de-fermat", "theorem/fermat-s-last-theorem" ] }, { "id": "ball-mathematical-recreations-1905/x-2898001b04", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 52", "location": "Some Arithmetical Questions", "latex": "A number is said to be perfect if it is equal to the sum of all its integral subdivisors. Thus the subdivisors of $6$ are $1$, $2$, and $3$; the sum of these is equal to $6$; hence $6$ is a perfect number.", "markdown": "A number is said to be perfect if it is equal to the sum of all its integral subdivisors. Thus the subdivisors of $6$ are $1$, $2$, and $3$; the sum of these is equal to $6$; hence $6$ is a perfect number.", "why": "A concrete example after a clear definition lets a learner check the idea on a small case before meeting the general formula.", "use": [ "lesson", "website" ], "concepts": [ "concept/perfect-number" ] }, { "id": "ball-mathematical-recreations-1905/x-1ba47bcbe6", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 48", "location": "Some Arithmetical Questions", "latex": "Thus only four weights are required, namely, $1$~lb., $3$~lbs., $3^2$~lbs., and $3^3$~lbs.", "markdown": "Thus only four weights are required, namely, $1$ lb., $3$ lbs., $3^2$ lbs., and $3^3$ lbs.", "why": "It states the answer to Bachet's problem in a form a learner can verify by hand against the weights 1 to 40.", "use": [ "lesson", "website" ], "concepts": [ "concept/bachet-s-weights-problem" ] }, { "id": "ball-mathematical-recreations-1905/x-db41f96b5a", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 49", "location": "Some Arithmetical Questions", "latex": "To determine the arrangement of the weights to weigh any given mass we have only to express the number of pounds in it as a number in the ternary scale of notation, except that in finding the successive digits we must make every remainder either $0$, $1$, or $-1$: to effect this a remainder $2$ must be written as $3-1$, that is, the quotient must be increased by unity, in which case the remainder is $-1$.", "markdown": "To determine the arrangement of the weights to weigh any given mass we have only to express the number of pounds in it as a number in the ternary scale of notation, except that in finding the successive digits we must make every remainder either $0$, $1$, or $-1$: to effect this a remainder $2$ must be written as $3-1$, that is, the quotient must be increased by unity, in which case the remainder is $-1$.", "why": "It gives a method a learner can apply to any mass, and the remainder rule shows why a plain base-three expansion is not enough.", "use": [ "lesson" ], "concepts": [ "concept/bachet-s-weights-problem", "concept/ternary-scale-of-notation" ] }, { "id": "ball-mathematical-recreations-1905/x-dd2bf3258e", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 59", "location": "Some Geometrical Questions", "latex": "I append two or three demonstrations, leading to obviously impossible results, which perhaps may amuse any one to whom they are new. I leave the discovery of the errors to the ingenuity of my readers.", "markdown": "I append two or three demonstrations, leading to obviously impossible results, which perhaps may amuse any one to whom they are new. I leave the discovery of the errors to the ingenuity of my readers.", "why": "It invites the learner to hunt for the flaw in each fallacy, which builds skill in checking every step of a proof.", "use": [ "lesson", "website" ], "concepts": [ "concept/geometrical-fallacy" ] }, { "id": "ball-mathematical-recreations-1905/x-d1882d4438", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 66", "location": "Some Geometrical Questions", "latex": "In fact proofs by superposition should be regarded with considerable distrust unless they are supplemented by mathematical reasoning.", "markdown": "In fact proofs by superposition should be regarded with considerable distrust unless they are supplemented by mathematical reasoning.", "why": "It warns the learner that a picture showing two shapes match is not a proof on its own.", "use": [ "lesson" ], "concepts": [ "concept/dissection-proof" ] }, { "id": "ball-mathematical-recreations-1905/x-b858fa4bb5", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 65", "location": "Some Geometrical Questions", "latex": "Rotate the lamina successively through two right angles about the diagonal $OB$ as axis and through two right angles about the side $OA$ as axis, and the required result will be attained.", "markdown": "Rotate the lamina successively through two right angles about the diagonal $OB$ as axis and through two right angles about the side $OA$ as axis, and the required result will be attained.", "why": "A short, surprising construction that shows how a sequence of simple rotations can produce a result that seems impossible.", "use": [ "lesson", "website" ], "concepts": [ "concept/geometrical-paradox" ] }, { "id": "ball-mathematical-recreations-1905/x-c94b525fc5", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 66", "location": "Some Geometrical Questions", "latex": "Hence the probability that a triangle can be constructed out of the three pieces into which the stick is broken would appear to be $\\frac{1}{2}$. This is not true, for actually the probability is $\\frac{1}{4}$.", "markdown": "Hence the probability that a triangle can be constructed out of the three pieces into which the stick is broken would appear to be $\\frac{1}{2}$. This is not true, for actually the probability is $\\frac{1}{4}$.", "why": "It shows a learner that an apparently sound probability argument can give a wrong answer, so each assumption must be checked.", "use": [ "lesson" ], "concepts": [ "concept/geometrical-paradox", "concept/probability" ] }, { "id": "ball-mathematical-recreations-1905/x-9303498897", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 71", "location": "Some Geometrical Questions", "latex": "All places whose heights above the mean sea level are equal are on the same level.", "markdown": "All places whose heights above the mean sea level are equal are on the same level.", "why": "It gives a plain, concrete definition of a contour-line that links geometry to the reading of a map.", "use": [ "lesson", "website" ], "concepts": [ "concept/contour-line" ] }, { "id": "ball-mathematical-recreations-1905/x-e3149f57f2", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 69", "location": "Some Geometrical Questions", "latex": "A proof of the proposition involves difficulties of a high order, which as yet have baffled all attempts to surmount them.", "markdown": "A proof of the proposition involves difficulties of a high order, which as yet have baffled all attempts to surmount them.", "why": "It tells the learner honestly that a plausible-sounding statement can remain unproved, which is an important point about what counts as established.", "use": [ "history", "lesson" ], "concepts": [ "theorem/four-colour-theorem" ] }, { "id": "ball-mathematical-recreations-1905/x-c03998c72c", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 73", "location": "Some Geometrical Questions", "latex": "The mathematical theory for a board of $9$ cells has been worked out completely, and there is no difficulty in extending it to one of $16$ cells: but the analysis is lengthy and not particularly interesting.", "markdown": "The mathematical theory for a board of $9$ cells has been worked out completely, and there is no difficulty in extending it to one of $16$ cells: but the analysis is lengthy and not particularly interesting.", "why": "It shows a learner that a finished analysis of a small board exists and that the larger case is a known extension, not an open problem.", "use": [ "history" ], "concepts": [ "concept/three-in-a-row" ] }, { "id": "ball-mathematical-recreations-1905/x-9c88e04306", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 74", "location": "Some Geometrical Questions", "latex": "Let $P$ be any point on a cubic. Let the tangent at $P$ cut the curve again in $Q$. Let the tangent at $Q$ cut the curve in $A$.", "markdown": "Let $P$ be any point on a cubic. Let the tangent at $P$ cut the curve again in $Q$. Let the tangent at $Q$ cut the curve in $A$.", "why": "It gives a concrete geometric recipe, built on repeated tangents, for placing counters so that many rows of three appear.", "use": [ "lesson" ], "concepts": [ "concept/tangent", "concept/three-in-a-row" ] }, { "id": "ball-mathematical-recreations-1905/x-2d0dac69f5", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 74", "location": "Some Geometrical Questions", "latex": "Sylvester stated that 9 counters can be\nplaced in 10 rows, each containing three counters; I do not\nknow how he placed them,", "markdown": "Sylvester stated that 9 counters can be placed in 10 rows, each containing three counters; I do not know how he placed them,", "why": "It is a striking example of a claim that beats the known bound, and the author openly says he cannot reproduce the arrangement.", "use": [ "history" ], "concepts": [ "concept/three-in-a-row", "person/james-joseph-sylvester" ] }, { "id": "ball-mathematical-recreations-1905/x-6c2ff588f1", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 75", "location": "Some Geometrical Questions", "latex": "To those who have never looked into the matter it may be\nsurprising that patterns formed by the use of square tiles (of\nwhich one-half bounded by a diagonal is white and the other\nhalf black) should be subject to mathematical analysis.", "markdown": "To those who have never looked into the matter it may be surprising that patterns formed by the use of square tiles (of which one-half bounded by a diagonal is white and the other half black) should be subject to mathematical analysis.", "why": "It invites a learner to see that everyday tiling patterns can be a real subject of mathematical analysis.", "use": [ "website" ], "concepts": [ "concept/tessellation" ] }, { "id": "ball-mathematical-recreations-1905/x-5209d90349", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 76", "location": "Some Geometrical Questions", "latex": "A cube has six faces, and if six\ncolours are chosen we can paint each face with a different\ncolour.", "markdown": "A cube has six faces, and if six colours are chosen we can paint each face with a different colour.", "why": "It sets up the counting of colourings of a cube from a simple, checkable starting point.", "use": [ "lesson" ], "concepts": [ "concept/colour-cube-problem", "concept/cube" ] }, { "id": "ball-mathematical-recreations-1905/x-37f9eaa338", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 80", "location": "Some Geometrical Questions", "latex": "Three beautiful ladies have for husbands three\nmen, who are as jealous as they are young, handsome, and\ngallant.", "markdown": "Three beautiful ladies have for husbands three men, who are as jealous as they are young, handsome, and gallant.", "why": "It states the classic constraint problem in a memorable way, though its wording is dated and should not be carried into a lesson unchanged.", "use": [ "website" ], "concepts": [ "concept/ferry-boat-problems", "person/alcuin" ] }, { "id": "ball-mathematical-recreations-1905/x-b03826d5b1", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 83", "location": "Some Geometrical Questions", "latex": "To obtain a solution we observe that we can cut a sheet of\npaper so that, when folded properly, it will make a model to\nscale of the room.", "markdown": "To obtain a solution we observe that we can cut a sheet of paper so that, when folded properly, it will make a model to scale of the room.", "why": "It teaches the unfolding idea that turns a shortest-route problem on a box into a straight line on a flat sheet.", "use": [ "lesson" ], "concepts": [ "concept/geodesic" ] }, { "id": "ball-mathematical-recreations-1905/x-c2ad4fe5c5", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 86", "location": "Some Geometrical Questions", "latex": "Suppose the pieces to be arranged originally in circular order, with two contiguous blank spaces, then we always move to the blank space for the time being that pair of coins which occupies the places next but one and next but two to the blank space on one assigned side of it.", "markdown": "Suppose the pieces to be arranged originally in circular order, with two contiguous blank spaces, then we always move to the blank space for the time being that pair of coins which occupies the places next but one and next but two to the blank space on one assigned side of it.", "why": "It states Tait's rule for choosing each move, so a learner can follow why each step in his solution is forced.", "use": [ "lesson" ], "concepts": [ "concept/tait-s-counter-problem" ] }, { "id": "ball-mathematical-recreations-1905/x-8893d4f7c9", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 90", "location": "Some Geometrical Questions", "latex": "Since there is only one cell on the board which is unoccupied, and since no diagonal moves and no backward moves are permitted, it follows that at each move not more than two pieces of either colour are capable of moving.", "markdown": "Since there is only one cell on the board which is unoccupied, and since no diagonal moves and no backward moves are permitted, it follows that at each move not more than two pieces of either colour are capable of moving.", "why": "It shows how restricting the moves to one direction and one empty cell makes a board puzzle analysable by hand.", "use": [ "lesson" ], "concepts": [ "concept/pawn-interchange-puzzle" ] }, { "id": "ball-mathematical-recreations-1905/x-0fd6b516f4", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 91", "location": "Some Geometrical Questions", "latex": "The solution is tolerably obvious. First, move the pieces from $a$ to $A$, from $b$ to $B$, from $c$ to $C$, and from $d$ to $D$.", "markdown": "The solution is tolerably obvious. First, move the pieces from $a$ to $A$, from $b$ to $B$, from $c$ to $C$, and from $d$ to $D$.", "why": "It opens the knights solution step by step, showing how a repeated pattern of moves produces the required exchange.", "use": [ "lesson" ], "concepts": [ "concept/guarini-s-knights-problem" ] }, { "id": "ball-mathematical-recreations-1905/x-f57be59906", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 92", "location": "Some Geometrical Questions", "latex": "Next suppose that the end $A$ is twisted once completely round (\\IE\\ through four right angles) before it is gummed to $B$, then a similar cut produces two interlaced rings.", "markdown": "Next suppose that the end $A$ is twisted once completely round ( through four right angles) before it is gummed to $B$, then a similar cut produces two interlaced rings.", "why": "It gives the striking experimental result that two full twists yield two interlaced rings, which a learner can test with paper.", "use": [ "website", "lesson" ], "concepts": [ "concept/paradromic-ring" ] }, { "id": "ball-mathematical-recreations-1905/x-17c84553e9", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 92", "location": "Some Geometrical Questions", "latex": "If any of my readers think that these results could be predicted off-hand, it may be interesting to them to see if they can predict correctly the effect of again cutting the rings formed in the second and third experiments down their middle lines in a manner similar to that above described.", "markdown": "If any of my readers think that these results could be predicted off-hand, it may be interesting to them to see if they can predict correctly the effect of again cutting the rings formed in the second and third experiments down their middle lines in a manner similar to that above described.", "why": "It invites the reader to predict an outcome before cutting, which makes it a good prompt for a hands-on exercise.", "use": [ "lesson", "website" ], "concepts": [ "concept/one-sided-surface", "concept/paradromic-ring" ] }, { "id": "ball-mathematical-recreations-1905/x-61954ae79c", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 89", "location": "Some Geometrical Questions", "latex": "Hence to interchange all the pieces will require $15 + (7 \\times 15)$ moves, that is, $120$ moves.", "markdown": "Hence to interchange all the pieces will require $15 + (7 \\times 15)$ moves, that is, $120$ moves.", "why": "It shows how the count of moves for a large board is built from the count for a smaller row, a useful pattern for counting arguments.", "use": [ "lesson" ], "concepts": [ "concept/pawn-interchange-puzzle" ] }, { "id": "ball-mathematical-recreations-1905/x-fc784b8214", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 73", "location": "Some Geometrical Questions", "latex": "Whoever first gets three (or any other assigned number) of his pieces in three adjacent cells and in a straight line wins.", "markdown": "Whoever first gets three (or any other assigned number) of his pieces in three adjacent cells and in a straight line wins.", "why": "Gives the exact winning rule of the game, the clearest starting point for the rest of the section.", "use": [ "lesson", "website" ], "concepts": [ "concept/three-in-a-row" ] }, { "id": "ball-mathematical-recreations-1905/x-e8b9f29785", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 75", "location": "Some Geometrical Questions", "latex": "Thus at present it is not possible to say what is the maximum number of rows of three which can be formed from $n$ counters placed on a plane.", "markdown": "Thus at present it is not possible to say what is the maximum number of rows of three which can be formed from $n$ counters placed on a plane.", "why": "Shows a learner an honest open question, which teaches that a maximum is not always known.", "use": [ "lesson", "history" ], "concepts": [ "concept/three-in-a-row" ] }, { "id": "ball-mathematical-recreations-1905/x-45b0ac6493", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 76", "location": "Some Geometrical Questions", "latex": "If more than two colours are used, the problems become increasingly difficult.", "markdown": "If more than two colours are used, the problems become increasingly difficult.", "why": "Tells the learner how adding colours changes the difficulty of a tiling puzzle.", "use": [ "lesson" ], "concepts": [ "concept/cross-fours", "concept/tessellation" ] }, { "id": "ball-mathematical-recreations-1905/x-04351237fc", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 77", "location": "Some Geometrical Questions", "latex": "Take any face of the cube $K$: it has four angles, and at each angle three colours meet.", "markdown": "Take any face of the cube $K$: it has four angles, and at each angle three colours meet.", "why": "Sets up the corner-based method that lets the colour-cube puzzle be solved by reasoning rather than trial.", "use": [ "lesson", "website" ], "concepts": [ "concept/colour-cube-problem", "concept/permutation" ] }, { "id": "ball-mathematical-recreations-1905/x-0638355da0", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 80", "location": "Some Geometrical Questions", "latex": "The construction and the initial arrangement ensure that at any one time there cannot be more than eight vehicles on the track.", "markdown": "The construction and the initial arrangement ensure that at any one time there cannot be more than eight vehicles on the track.", "why": "Shows how the rules of a shunting puzzle restrict the possible states, which is the key constraint in the problem.", "use": [ "lesson" ], "concepts": [ "concept/shunting-problems" ] }, { "id": "ball-mathematical-recreations-1905/x-e5a8255474", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 81", "location": "Some Geometrical Questions", "latex": "Let $y$ denote the number of passages from one bank to the other which will be necessary.", "markdown": "Let $y$ denote the number of passages from one bank to the other which will be necessary.", "why": "Introduces the notation for counting crossings and frames the question of a minimum number of passages.", "use": [ "lesson" ], "concepts": [ "concept/ferry-boat-problems" ] }, { "id": "ball-mathematical-recreations-1905/x-03cf7a3e99", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 84", "location": "Some Geometrical Questions", "latex": "Thus the problem is reduced to finding the way of cutting out the paper which gives the shortest route of the kind.", "markdown": "Thus the problem is reduced to finding the way of cutting out the paper which gives the shortest route of the kind.", "why": "Explains the unfolding idea behind the wasp-and-fly puzzle, a method a learner can reuse on other surfaces.", "use": [ "lesson", "website" ], "concepts": [ "concept/geodesic" ] }, { "id": "ball-mathematical-recreations-1905/x-af1df12a95", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 108", "location": "Some Mechanical Questions", "latex": "Hence the forces that act on the machine and are brought into play by the various parts may be altered in different proportions, and thus the machine may be incapable of producing results similar to those which can be produced by the model.", "markdown": "Hence the forces that act on the machine and are brought into play by the various parts may be altered in different proportions, and thus the machine may be incapable of producing results similar to those which can be produced by the model.", "why": "It explains the practical consequence of scaling, showing why a model can mislead a designer.", "use": [ "lesson", "history" ], "concepts": [ "concept/scaling-of-a-model", "concept/similarity", "quantity/force" ] }, { "id": "ball-mathematical-recreations-1905/x-ada2b6c92e", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 110", "location": "Some Mechanical Questions", "latex": "But as the velocity of the boat increases, a time will arrive when the pressure of the wind is only just able to balance the resisting force which is caused by the sail moving through the air.", "markdown": "But as the velocity of the boat increases, a time will arrive when the pressure of the wind is only just able to balance the resisting force which is caused by the sail moving through the air.", "why": "It describes steady motion in ordinary words, so a learner can picture why the speed levels off.", "use": [ "lesson" ], "concepts": [ "concept/pressure", "concept/steady-motion", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/x-9641f26bef", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 113", "location": "Some Mechanical Questions", "latex": "Hence the usual movements of the crew in the boat do not sensibly move the centre of gravity of themselves and the boat, but this does not apply to an impulsive movement, and if the crew in making a jerk move their centre of gravity towards the bow $n$ times more rapidly than it returns after the jerk, then the boat is impelled forwards at least $n$ times more than backwards: hence on the whole the motion is forwards", "markdown": "Hence the usual movements of the crew in the boat do not sensibly move the centre of gravity of themselves and the boat, but this does not apply to an impulsive movement, and if the crew in making a jerk move their centre of gravity towards the bow $n$ times more rapidly than it returns after the jerk, then the boat is impelled forwards at least $n$ times more than backwards: hence on the whole the motion is forwards", "why": "It shows how a difference in speeds of motion, not total motion, drives the boat, which is a clear lesson in centre of gravity and friction.", "use": [ "lesson", "website" ], "concepts": [ "concept/boat-moved-by-a-rope", "concept/centre-of-gravity", "concept/friction" ] }, { "id": "ball-mathematical-recreations-1905/x-d549719745", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 99", "location": "Some Mechanical Questions", "latex": "Thus, if a number of dominoes or draughts are arranged in a vertical pile, a sharp horizontal blow on one of those near the bottom will send it out of the pile, and those above will merely drop down to take its place---in fact they have not time to change their relative positions before there is sufficient space for them to drop vertically as if they were a solid body.", "markdown": "Thus, if a number of dominoes or draughts are arranged in a vertical pile, a sharp horizontal blow on one of those near the bottom will send it out of the pile, and those above will merely drop down to take its place---in fact they have not time to change their relative positions before there is sufficient space for them to drop vertically as if they were a solid body.", "why": "A concrete picture of inertia that a learner can test at home with a stack of coins or dominoes.", "use": [ "lesson", "website" ], "concepts": [ "concept/inertia", "law/first-law-of-motion" ] }, { "id": "ball-mathematical-recreations-1905/x-6c415a86c0", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 102", "location": "Some Mechanical Questions", "latex": "In other words, if in order to cause a displacement work has to be done against the forces acting on the body, then for that displacement the equilibrium is stable, while if the forces do work the equilibrium is unstable.", "markdown": "In other words, if in order to cause a displacement work has to be done against the forces acting on the body, then for that displacement the equilibrium is stable, while if the forces do work the equilibrium is unstable.", "why": "It links the stability of a body to the work needed to displace it, which is a useful test a learner can apply to new cases.", "use": [ "lesson" ], "concepts": [ "concept/centre-of-gravity", "concept/stability-of-equilibrium", "concept/work" ] }, { "id": "ball-mathematical-recreations-1905/x-a23522339c", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 96", "location": "Some Mechanical Questions", "latex": "To establish this, Zeno argued that when Achilles had gone the $1000$ yards, the tortoise would still be $100$ yards in front of him; by the time he had covered these $100$ yards, it would still be $10$ yards in front of him; and so on for ever.", "markdown": "To establish this, Zeno argued that when Achilles had gone the $1000$ yards, the tortoise would still be $100$ yards in front of him; by the time he had covered these $100$ yards, it would still be $10$ yards in front of him; and so on for ever.", "why": "It shows the reasoning behind a famous paradox step by step, so learners can see exactly where the infinite regress is set up.", "use": [ "lesson", "website" ], "concepts": [ "concept/achilles-and-the-tortoise", "concept/zeno-s-paradoxes" ] }, { "id": "ball-mathematical-recreations-1905/x-c18bc7c6df", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 95", "location": "Some Mechanical Questions", "latex": "So naturalists observe, a flea hath smaller fleas that on him prey.", "markdown": "So naturalists observe, a flea hath smaller fleas that on him prey.", "why": "A memorable old verse that makes the idea of an infinite regress vivid for a general reader.", "use": [ "website", "history" ], "concepts": [ "concept/zeno-s-paradoxes", "person/jonathan-swift" ] }, { "id": "ball-mathematical-recreations-1905/x-76e1dc0b30", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 102", "location": "Some Mechanical Questions", "latex": "Montucla\\index{Montucla} says that in his time it was not uncommon to see boxes of tin soldiers mounted on lead hemispheres, and when the lid of the box was taken off the whole regiment sprang to attention.", "markdown": "Montucla says that in his time it was not uncommon to see boxes of tin soldiers mounted on lead hemispheres, and when the lid of the box was taken off the whole regiment sprang to attention.", "why": "A lively historical picture of stable and unstable equilibrium that makes the principle easy to remember.", "use": [ "history", "website" ], "concepts": [ "concept/stability-of-equilibrium", "person/jean-tienne-montucla" ] }, { "id": "ball-mathematical-recreations-1905/x-7cdcfe8e69", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 98", "location": "Some Mechanical Questions", "latex": "Probably the meaning of the law is best expressed in Clifford's\\index{Clifford} phrase, that force is ``the description of a certain kind of motion''---in other words it is not an entity but merely a convenient way of stating, without circumlocution, that a certain kind of motion is observed.", "markdown": "Probably the meaning of the law is best expressed in Clifford’s phrase, that force is “the description of a certain kind of motion”---in other words it is not an entity but merely a convenient way of stating, without circumlocution, that a certain kind of motion is observed.", "why": "It gives learners a clear, memorable way to think of force as a description of motion rather than a thing.", "use": [ "lesson", "history" ], "concepts": [ "law/first-law-of-motion", "person/william-kingdon-clifford", "quantity/force" ] }, { "id": "ball-mathematical-recreations-1905/x-60f2bb43cc", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 101", "location": "Some Mechanical Questions", "latex": "A given agent in a given time can do only a definite amount of work. This is illustrated by the fact that although, by means of a rigid lever and a fixed fulcrum, any force however small may be caused to move any mass however large, yet what is gained in power is lost in speed---as the popular phrase runs.", "markdown": "A given agent in a given time can do only a definite amount of work. This is illustrated by the fact that although, by means of a rigid lever and a fixed fulcrum, any force however small may be caused to move any mass however large, yet what is gained in power is lost in speed---as the popular phrase runs.", "why": "It explains the conservation of work through levers, showing that gaining force costs speed.", "use": [ "lesson" ], "concepts": [ "concept/energy", "concept/work", "instrument/lever" ] }, { "id": "ball-mathematical-recreations-1905/x-4369f8bda4", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 95", "location": "Some Mechanical Questions", "latex": "The assertion was that if Achilles ran ten times as fast as a tortoise, yet if the tortoise had (say) $1000$ yards start it could never be overtaken.", "markdown": "The assertion was that if Achilles ran ten times as fast as a tortoise, yet if the tortoise had (say) $1000$ yards start it could never be overtaken.", "why": "It states Zeno's paradox clearly so that learners can set up the infinite-sum resolution themselves.", "use": [ "lesson", "website" ], "concepts": [ "concept/achilles-and-the-tortoise", "concept/zeno-s-paradoxes" ] }, { "id": "ball-mathematical-recreations-1905/x-1b8bf86333", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 102", "location": "Some Mechanical Questions", "latex": "Such a bottle is made of thin glass or varnished paper fixed to the plane surface of a solid hemisphere or smaller segment of a sphere.", "markdown": "Such a bottle is made of thin glass or varnished paper fixed to the plane surface of a solid hemisphere or smaller segment of a sphere.", "why": "It describes the magic bottle's construction, the first step in explaining its stability through the centre of gravity.", "use": [ "lesson", "website" ], "concepts": [ "concept/centre-of-gravity", "concept/magic-bottle", "concept/stability-of-equilibrium" ] }, { "id": "ball-mathematical-recreations-1905/x-148e1fbd2f", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 107", "location": "Some Mechanical Questions", "latex": "Of course it is only in isolated systems that the total amount of energy is constant, and, if a source of external energy can be obtained from which energy is continually introduced into the system, perpetual motion is, in a sense, possible; though even here materials would ultimately wear out.", "markdown": "Of course it is only in isolated systems that the total amount of energy is constant, and, if a source of external energy can be obtained from which energy is continually introduced into the system, perpetual motion is, in a sense, possible; though even here materials would ultimately wear out.", "why": "It warns learners that energy conservation forbids perpetual motion only in isolated systems, a common misreading.", "use": [ "lesson" ], "concepts": [ "concept/energy", "concept/isolated-system", "concept/perpetual-motion" ] }, { "id": "ball-mathematical-recreations-1905/x-0b33997537", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 101", "location": "Some Mechanical Questions", "latex": "Montucla\\index{Montucla} calculated the mass of the earth and, assuming that a man could work incessantly at the rate of $116$ foot-lbs.\\ per second, which is a very high estimate, he found that it would take over three billion centuries, \\IE\\ $3 \\times 10^{14}$ years, before a mass equal to that of the earth was moved as much as one inch against gravity at the surface of the earth", "markdown": "Montucla calculated the mass of the earth and, assuming that a man could work incessantly at the rate of $116$ foot-lbs. per second, which is a very high estimate, he found that it would take over three billion centuries, $3 \\times 10^{14}$ years, before a mass equal to that of the earth was moved as much as one inch against gravity at the surface of the earth", "why": "A vivid, quantified illustration of how work and the lever principle play out at a huge scale.", "use": [ "history", "website" ], "concepts": [ "concept/work", "instrument/lever", "person/jean-tienne-montucla", "unit/foot-pound" ] }, { "id": "ball-mathematical-recreations-1905/x-b633954eab", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 108", "location": "Some Mechanical Questions", "latex": "If all the parts of a model are magnified in the same proportion, say $m$, and if thereby a line in it is increased in the ratio $m:1$, then the areas and volumes in it will be increased respectively in the ratios $m^2:1$ and $m^3:1$.", "markdown": "If all the parts of a model are magnified in the same proportion, say $m$, and if thereby a line in it is increased in the ratio $m:1$, then the areas and volumes in it will be increased respectively in the ratios $m^2:1$ and $m^3:1$.", "why": "It states the core scaling rule in one sentence, so a learner can see why areas and volumes grow as powers of the linear factor.", "use": [ "lesson", "website" ], "concepts": [ "concept/corresponding-linear-dimensions", "concept/scaling-of-a-model", "quantity/area", "quantity/volume" ] }, { "id": "ball-mathematical-recreations-1905/x-f676ad7d4a", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 108", "location": "Some Mechanical Questions", "latex": "For example, if the side of a cube is doubled then a face of it will be increased in the ratio $4: 1$ and its volume will be increased in the ratio $8:1$.", "markdown": "For example, if the side of a cube is doubled then a face of it will be increased in the ratio $4: 1$ and its volume will be increased in the ratio $8:1$.", "why": "The doubled cube makes the powers 4 and 8 concrete, which a learner can check by counting faces and small cubes.", "use": [ "lesson" ], "concepts": [ "concept/cube", "concept/face", "concept/scaling-of-a-model", "quantity/area", "quantity/volume" ] }, { "id": "ball-mathematical-recreations-1905/x-a54bfde0a1", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 108", "location": "Some Mechanical Questions", "latex": "Again, if the linear dimensions of a man of height $5$~ft.\\ $10$~in.\\ were increased by one-seventh his height would become $6$~ft.\\ $8$~in., but his weight would be increased in the ratio $512: 343$ (\\IE\\ about half as much again), while the cross sections of his legs, which would have to bear this weight, would be increased only in the ratio $64:49$; thus in some respects he would be less efficient than before.", "markdown": "Again, if the linear dimensions of a man of height $5$ ft. $10$ in. were increased by one-seventh his height would become $6$ ft. $8$ in., but his weight would be increased in the ratio $512: 343$ ( about half as much again), while the cross sections of his legs, which would have to bear this weight, would be increased only in the ratio $64:49$; thus in some respects he would be less efficient than before.", "why": "A worked numerical example of weight growing as the cube while leg cross-sections grow as the square, which makes the scaling argument vivid.", "use": [ "lesson", "history" ], "concepts": [ "concept/cross-section", "concept/scaling-of-a-model", "concept/weight" ] }, { "id": "ball-mathematical-recreations-1905/x-7caff6710f", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "Hence, if the wind makes the same angle $\\alpha$ abaft the beam that the sail makes with the keel, the velocity of the boat will be greater than the velocity of the wind.", "markdown": "Hence, if the wind makes the same angle $\\alpha$ abaft the beam that the sail makes with the keel, the velocity of the boat will be greater than the velocity of the wind.", "why": "It states the surprising conclusion of the sailing argument in plain words, so a learner knows what the calculation is meant to show.", "use": [ "website", "lesson" ], "concepts": [ "concept/maximum", "concept/sailing-faster-than-the-wind", "quantity/angle", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/x-f57ba0cde5", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 113", "location": "Some Mechanical Questions", "latex": "The chief cause for this result seems to be that the friction between the boat and the water retards all relative motion, but is not great enough to materially affect motion caused by a sufficiently big impulse.", "markdown": "The chief cause for this result seems to be that the friction between the boat and the water retards all relative motion, but is not great enough to materially affect motion caused by a sufficiently big impulse.", "why": "It gives the physical reason a rope-propelled boat moves forward, tying friction and impulse together in one sentence.", "use": [ "lesson" ], "concepts": [ "concept/boat-moved-by-a-rope", "concept/friction", "concept/impulse" ] }, { "id": "ball-mathematical-recreations-1905/x-a51b2fc84e", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 113", "location": "Some Mechanical Questions", "latex": "Thus, if a current of air is moving in a tube, the pressure on the sides of the tube is less than when the air is at rest---and the quicker the air moves the smaller is the pressure.", "markdown": "Thus, if a current of air is moving in a tube, the pressure on the sides of the tube is less than when the air is at rest---and the quicker the air moves the smaller is the pressure.", "why": "It states Hauksbee's law as a plain observation a learner can test by blowing through a tube.", "use": [ "lesson", "history" ], "concepts": [ "concept/pressure", "law/hauksbee-s-law" ] }, { "id": "ball-mathematical-recreations-1905/x-ad73525262", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 114", "location": "Some Mechanical Questions", "latex": "If anyone blows steadily through the tube so formed, the paper will be sucked in instead of being blown out.", "markdown": "If anyone blows steadily through the tube so formed, the paper will be sucked in instead of being blown out.", "why": "A simple household experiment that makes the counter-intuitive pressure drop in moving air visible.", "use": [ "lesson", "website" ], "concepts": [ "concept/motion-in-fluids", "law/hauksbee-s-law" ] }, { "id": "ball-mathematical-recreations-1905/x-c1eee91df8", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 136", "location": "Some Miscellaneous Questions", "latex": "The first of these is to place eight queens on a chess-board so as to command the fewest possible squares.", "markdown": "The first of these is to place eight queens on a chess-board so as to command the fewest possible squares.", "why": "It states the eight-queens variant as a clear optimisation question that a learner can grasp at once.", "use": [ "website" ], "concepts": [ "concept/chess-piece-placement-problem" ] }, { "id": "ball-mathematical-recreations-1905/x-e33cdf3a22", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 139", "location": "Some Miscellaneous Questions", "latex": "Let $a$ stand for $a_1$, or $a_2$, $b$ for $b_1$ or $b_2$, and so on.", "markdown": "Let $a$ stand for $a_1$, or $a_2$, $b$ for $b_1$ or $b_2$, and so on.", "why": "It shows the labelling step that starts Frost's method, so a learner can follow the notation.", "use": [ "lesson" ], "concepts": [ "concept/complementary-suffixes", "method/frost-s-method" ] }, { "id": "ball-mathematical-recreations-1905/x-165be835b9", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 139", "location": "Some Miscellaneous Questions", "latex": "Thus there are seven possible triads, such as $abc$, $ade$, $afg$, $bdf$, $beg$, $cdg$, and $cef$.", "markdown": "Thus there are seven possible triads, such as $abc$, $ade$, $afg$, $bdf$, $beg$, $cdg$, and $cef$.", "why": "It lists concrete examples of triads so a learner can check the counting for themselves.", "use": [ "lesson" ], "concepts": [ "concept/triad", "method/frost-s-method" ] }, { "id": "ball-mathematical-recreations-1905/x-d25436e792", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 137", "location": "Some Miscellaneous Questions", "latex": "It has been asserted that, if $k = 2$, the number of ways in which two kings can be placed on a board so that they may not occupy adjacent squares is $\\frac{1}{2}(n-1)(n-2) (n^2 + 3n-2)$.", "markdown": "It has been asserted that, if $k = 2$, the number of ways in which two kings can be placed on a board so that they may not occupy adjacent squares is $\\frac{1}{2}(n-1)(n-2) (n^2 + 3n-2)$.", "why": "It records a formula offered as an assertion, so a reader can see it is a claim from the literature and not a result given in this book.", "use": [ "history" ], "concepts": [ "concept/chess-piece-placement-problem" ] }, { "id": "ball-mathematical-recreations-1905/x-2431185198", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 121", "location": "Some Miscellaneous Questions", "latex": "Now a cyclical permutation of $n$ letters is equivalent to\n$n-1$ simple interchanges; accordingly an odd cyclical permutation\nis equivalent to an even number of simple interchanges.", "markdown": "Now a cyclical permutation of $n$ letters is equivalent to $n-1$ simple interchanges; accordingly an odd cyclical permutation is equivalent to an even number of simple interchanges.", "why": "It gives the step that turns the sliding-block puzzle into a parity test, which a learner can follow with a small case.", "use": [ "lesson" ], "concepts": [ "concept/cyclical-permutation", "concept/fifteen-puzzle" ] }, { "id": "ball-mathematical-recreations-1905/x-f11c553f38", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 124", "location": "Some Miscellaneous Questions", "latex": "Hence, if it requires $x$ transfers of simple discs to move a\ntower of $n-1$ discs, then it will require $2x +1$ separate\ntransfers of single discs to move a tower of $n$ discs.", "markdown": "Hence, if it requires $x$ transfers of simple discs to move a tower of $n-1$ discs, then it will require $2x +1$ separate transfers of single discs to move a tower of $n$ discs.", "why": "It states the recursive step of the Tower of Hanoi clearly enough that a learner can reproduce the count for three or four discs.", "use": [ "lesson" ], "concepts": [ "concept/tower-of-hanoi" ] }, { "id": "ball-mathematical-recreations-1905/x-0803ab2792", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 125", "location": "Some Miscellaneous Questions", "latex": "Proceeding in this way we see\nthat with a tower of $n$ discs it will require $2^n-1$ transfers\nof single discs to effect the complete transfer.", "markdown": "Proceeding in this way we see that with a tower of $n$ discs it will require $2^n-1$ transfers of single discs to effect the complete transfer.", "why": "It gives the closed form 2^n - 1 that the recursion produces, showing how a pattern becomes a formula.", "use": [ "lesson" ], "concepts": [ "concept/tower-of-hanoi" ] }, { "id": "ball-mathematical-recreations-1905/x-cc3482bea4", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 129", "location": "Some Miscellaneous Questions", "latex": "Denote the rings which are on the bar by the digits $1$ or $0$\nalternately, reckoning from left to right, and denote a ring\nwhich is off the bar by the digit assigned to that ring\non the bar which is nearest to it on the left of it, or by\na $0$ if there is no ring to the left of it.", "markdown": "Denote the rings which are on the bar by the digits $1$ or $0$ alternately, reckoning from left to right, and denote a ring which is off the bar by the digit assigned to that ring on the bar which is nearest to it on the left of it, or by a $0$ if there is no ring to the left of it.", "why": "It shows how a physical puzzle is turned into binary counting, so that each move is adding or subtracting one.", "use": [ "lesson", "website" ], "concepts": [ "concept/binary-scale", "concept/chinese-rings" ] }, { "id": "ball-mathematical-recreations-1905/x-fa71bd947a", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 132", "location": "Some Miscellaneous Questions", "latex": "In the case of an ordinary chess-board the determinant\nis of the $8$th order, and therefore contains $8!$, that is, $40320$\nterms, so that it would be out of the question to use this\nmethod for the usual chess-board of $64$ cells or for a board of\nlarger size unless some way of picking out the required terms\ncould be discovered.", "markdown": "In the case of an ordinary chess-board the determinant is of the $8$th order, and therefore contains $8!$, that is, $40320$ terms, so that it would be out of the question to use this method for the usual chess-board of $64$ cells or for a board of larger size unless some way of picking out the required terms could be discovered.", "why": "It is a plain warning that a correct method can still be impractical, which is a useful habit to teach.", "use": [ "lesson" ], "concepts": [ "concept/chess-piece-placement-problem", "concept/determinant" ] }, { "id": "ball-mathematical-recreations-1905/x-a87f55e0e6", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 125", "location": "Some Miscellaneous Questions", "latex": "Would that English writers were\nin the habit of inventing equally interesting origins for the\npuzzles they produce!", "markdown": "Would that English writers were in the habit of inventing equally interesting origins for the puzzles they produce!", "why": "It is a lively old turn of phrase that gives the Tower of Hanoi its legend and suits a history reading.", "use": [ "history", "website" ], "concepts": [ "concept/tower-of-hanoi" ] }, { "id": "ball-mathematical-recreations-1905/x-5180c16c18", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 138", "location": "Some Miscellaneous Questions", "latex": "The problem is to dispose them so that for seven consecutive days no girl will walk with any of her school-fellows more than once.", "markdown": "The problem is to dispose them so that for seven consecutive days no girl will walk with any of her school-fellows more than once.", "why": "It states Kirkman's schoolgirl problem in its clearest form, so a learner can see exactly what must be arranged.", "use": [ "lesson", "website" ], "concepts": [ "concept/kirkman-s-schoolgirl-problem" ] }, { "id": "ball-mathematical-recreations-1905/x-cb308bebfa", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 139", "location": "Some Miscellaneous Questions", "latex": "The suffixes $1$ and $2$ are called complementary.", "markdown": "The suffixes $1$ and $2$ are called complementary.", "why": "It names the device that Frost's method relies on, so a learner can follow how the letters are labelled.", "use": [ "lesson" ], "concepts": [ "concept/complementary-suffixes", "method/frost-s-method" ] }, { "id": "ball-mathematical-recreations-1905/x-44e48856ae", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 140", "location": "Some Miscellaneous Questions", "latex": "Then, if the suffixes are permuted cyclically, we obtain six other arrangements which satisfy the conditions of the problem:", "markdown": "Then, if the suffixes are permuted cyclically, we obtain six other arrangements which satisfy the conditions of the problem:", "why": "It shows in one sentence how a single day's arrangement generates the rest of the week by cyclical permutation.", "use": [ "lesson" ], "concepts": [ "concept/cyclical-permutation", "method/anstice-s-method" ] }, { "id": "ball-mathematical-recreations-1905/x-94fa2d7c16", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 136", "location": "Some Miscellaneous Questions", "latex": "Is it possible to place the eight queens so as to leave more than eleven cells out of check? I have never succeeded in doing so, nor in showing that it is impossible to do it.", "markdown": "Is it possible to place the eight queens so as to leave more than eleven cells out of check? I have never succeeded in doing so, nor in showing that it is impossible to do it.", "why": "It presents an honest open question, which shows learners that some board problems remain unsolved.", "use": [ "website", "history" ], "concepts": [ "concept/board-covering", "concept/chess-piece-placement-problem" ] }, { "id": "ball-mathematical-recreations-1905/x-d04201ef23", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 145", "location": "Some Miscellaneous Questions", "latex": "Suppose for instance that a pack of $n$ cards is shuffled, as is not unusual, by placing the second card on the first, the third below these, the fourth above them, and so on.", "markdown": "Suppose for instance that a pack of $n$ cards is shuffled, as is not unusual, by placing the second card on the first, the third below these, the fourth above them, and so on.", "why": "It gives a concrete shuffling rule that makes the abstract idea of a repeated permutation easy to picture.", "use": [ "lesson", "website" ], "concepts": [ "method/card-shuffling", "person/monge" ] }, { "id": "ball-mathematical-recreations-1905/x-330ede2a98", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 147", "location": "Some Miscellaneous Questions", "latex": "Let the number $n$ of the pack be divided into $p, q, r, \\ldots$ such cycles, whose sum is $n$; then the \\textsc{l.c.m.} of $p, q, r, \\ldots$ is the utmost number of shufflings necessary before all the cards will be brought back to their original places.", "markdown": "Let the number $n$ of the pack be divided into $p, q, r, \\ldots$ such cycles, whose sum is $n$; then the l.c.m. of $p, q, r, \\ldots$ is the utmost number of shufflings necessary before all the cards will be brought back to their original places.", "why": "It links cycle lengths to a least common multiple, showing learners why that arithmetic governs the shuffling problem.", "use": [ "lesson" ], "concepts": [ "concept/cycle-of-operations", "concept/lowest-common-multiple", "theorem/hudson-s-shuffling-theorem" ] }, { "id": "ball-mathematical-recreations-1905/x-ab6f0b9c3b", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 149", "location": "Some Miscellaneous Questions", "latex": "For example, if we are told that in figure~i the card is in the third row, it must be either $9$, $10$, $11$, or $12$: hence, if we know in which row of figure~ii it lies, it is determined.", "markdown": "For example, if we are told that in figure i the card is in the third row, it must be either $9$, $10$, $11$, or $12$: hence, if we know in which row of figure ii it lies, it is determined.", "why": "It explains the row-and-column card trick with a concrete example that a learner can check by hand.", "use": [ "lesson", "website" ], "concepts": [ "concept/arrangement", "concept/remainder" ] }, { "id": "ball-mathematical-recreations-1905/x-09005ce4a5", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 149", "location": "Some Miscellaneous Questions", "latex": "This depends on the fact that the number of homogeneous products of two dimensions which can be formed out of four things is $10$. Hence the homogeneous products of two dimensions formed out of four things can be used to define ten things.", "markdown": "This depends on the fact that the number of homogeneous products of two dimensions which can be formed out of four things is $10$. Hence the homogeneous products of two dimensions formed out of four things can be used to define ten things.", "why": "It explains in one step why a trick with ten pairs of cards works, by linking it to counting products.", "use": [ "lesson" ], "concepts": [ "concept/homogeneous-polynomial", "method/pairs-of-cards-trick" ] }, { "id": "ball-mathematical-recreations-1905/x-807193dcfb", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 152", "location": "Some Miscellaneous Questions", "latex": "Request a spectator to note a card, and remember in which pile it is. After finishing the deal, ask in which pile the card is.", "markdown": "Request a spectator to note a card, and remember in which pile it is. After finishing the deal, ask in which pile the card is.", "why": "It gives the opening moves of the three-pile trick, which a learner can act out with real cards.", "use": [ "lesson" ], "concepts": [ "method/three-pile-problem" ] }, { "id": "ball-mathematical-recreations-1905/x-671f35d47a", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 153", "location": "Some Miscellaneous Questions", "latex": "The reason is that after the first deal you know it is one of sixty-four cards. In the next deal these sixty-four cards are distributed equally over the four piles, and therefore, if you know in which pile it is, you will know that it is one of sixteen cards.", "markdown": "The reason is that after the first deal you know it is one of sixty-four cards. In the next deal these sixty-four cards are distributed equally over the four piles, and therefore, if you know in which pile it is, you will know that it is one of sixteen cards.", "why": "It shows the halving logic behind Gergonne's pile method, one deal at a time, in a form a learner can follow.", "use": [ "lesson", "website" ], "concepts": [ "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/x-889d7fc46b", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 155", "location": "Some Miscellaneous Questions", "latex": "Hence, if $n-1$ is expressed in the ternary scale of notation, $x$, $y$, $z$ will be determined, and therefore $a$, $b$, $c$ will be known.", "markdown": "Hence, if $n-1$ is expressed in the ternary scale of notation, $x$, $y$, $z$ will be determined, and therefore $a$, $b$, $c$ will be known.", "why": "It links the pile positions to base-three notation, a link a learner can check by hand.", "use": [ "lesson" ], "concepts": [ "concept/ternary-scale-of-notation", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/x-6eab8eab08", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 156", "location": "Some Miscellaneous Questions", "latex": "If the $k$th card has the number $k$ on it---which event is called a \\emph{hit}---the player takes up the card and begins counting afresh.", "markdown": "If the $k$th card has the number $k$ on it---which event is called a *hit*---the player takes up the card and begins counting afresh.", "why": "It defines the key term of the mouse trap game precisely, so the rules can be followed step by step.", "use": [ "lesson", "website" ], "concepts": [ "concept/mouse-trap-game" ] }, { "id": "ball-mathematical-recreations-1905/x-80b7a79e49", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 151", "location": "Some Miscellaneous Questions", "latex": "I believe that these arrangements by sentences are known, but I am not aware who invented them", "markdown": "I believe that these arrangements by sentences are known, but I am not aware who invented them", "why": "It is the author's candid statement of what he does not know, a useful model of honest attribution.", "use": [ "history" ], "concepts": [ "method/pairs-of-cards-trick" ] }, { "id": "ball-mathematical-recreations-1905/x-eea472b2ec", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 172", "location": "Magic Squares", "latex": "For simplicity I shall apply this method to construct a magic square of only the sixth order, though an exactly similar method will apply to any even square of an order higher than the second.", "markdown": "For simplicity I shall apply this method to construct a magic square of only the sixth order, though an exactly similar method will apply to any even square of an order higher than the second.", "why": "It tells the learner that the procedure is shown on one case (order six) and that the same steps work for any even order above two.", "use": [ "lesson" ], "concepts": [ "concept/magic-square", "concept/order-of-a-magic-square", "method/constructing-an-even-magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-73112b87b1", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 171", "location": "Magic Squares", "latex": "Following the analogy of the notation used above, two numbers which are equidistant from the ends of the series $1,2,3,\\dotsc,n$ are said to be \\emph{complementary}.", "markdown": "Following the analogy of the notation used above, two numbers which are equidistant from the ends of the series $1,2,3,\\dotsc,n$ are said to be *complementary*.", "why": "It gives the exact meaning of 'complementary' that the construction relies on, so the learner can check each placement against it.", "use": [ "lesson" ], "concepts": [ "concept/complementary-numbers" ] }, { "id": "ball-mathematical-recreations-1905/x-54ef770ccc", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 171", "location": "Magic Squares", "latex": "we begin by constructing two subsidiary squares, one of the unit-digits, $1,2,3,\\dotsc,n$, and the other of the radix-digits $0,n,2n,\\dotsc,(n-1)n$.", "markdown": "we begin by constructing two subsidiary squares, one of the unit-digits, $1,2,3,\\dotsc,n$, and the other of the radix-digits $0,n,2n,\\dotsc,(n-1)n$.", "why": "It names the two building blocks of the method and shows that a magic square is made by adding them cell by cell.", "use": [ "lesson" ], "concepts": [ "concept/digit", "concept/subsidiary-square", "method/constructing-an-even-magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-34bdea7dce", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 171", "location": "Magic Squares", "latex": "I do not know to whom the modification is due.", "markdown": "I do not know to whom the modification is due.", "why": "It is the author's own candid admission about the origin of the enunciation he uses, a useful model of honest attribution for a reader.", "use": [ "history" ], "concepts": [ "method/constructing-an-even-magic-square", "person/de-la-hire" ] }, { "id": "ball-mathematical-recreations-1905/x-899711b864", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 158", "location": "Magic Squares", "latex": "I confine my account to such magic squares, that is, to squares formed with consecutive integers, from $1$ upwards.", "markdown": "I confine my account to such magic squares, that is, to squares formed with consecutive integers, from $1$ upwards.", "why": "It fixes the scope: only squares built from the consecutive integers 1 upwards are discussed.", "use": [ "lesson" ], "concepts": [ "concept/magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-9a01ff5a8d", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 158", "location": "Magic Squares", "latex": "If the integers are the consecutive numbers from $1$ to $n^2$ the square is said to be of the $n$th order, and it is easily seen that in this case the sum of the numbers in any row, column, or diagonal is equal to $\\frac{1}{2}n(n^2 +1)$: this number may be denoted by $N$.", "markdown": "If the integers are the consecutive numbers from $1$ to $n^2$ the square is said to be of the $n$th order, and it is easily seen that in this case the sum of the numbers in any row, column, or diagonal is equal to $\\frac{1}{2}n(n^2 +1)$: this number may be denoted by $N$.", "why": "It gives the order and the magic sum in one statement, which a learner can check on a small square.", "use": [ "lesson" ], "concepts": [ "concept/order-of-a-magic-square", "quantity/magic-sum" ] }, { "id": "ball-mathematical-recreations-1905/x-991736fa82", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 158", "location": "Magic Squares", "latex": "The formation of these squares is an old amusement, and in times when mystical philosophical ideas were associated with particular numbers it was natural that such arrangements should be deemed to possess magical properties.", "markdown": "The formation of these squares is an old amusement, and in times when mystical philosophical ideas were associated with particular numbers it was natural that such arrangements should be deemed to possess magical properties.", "why": "It places magic squares in their historical setting as a recreation shaped by the beliefs of the time.", "use": [ "history", "website" ], "concepts": [ "concept/magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-7b204a6df8", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 159", "location": "Magic Squares", "latex": "Magic squares of an odd order were constructed in India before the Christian era according to a law of formation which is explained hereafter.", "markdown": "Magic squares of an odd order were constructed in India before the Christian era according to a law of formation which is explained hereafter.", "why": "It dates the odd-order construction to ancient India, a historical point a teacher can share.", "use": [ "history" ], "concepts": [ "concept/magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-3c8f2df7d7", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 159", "location": "Magic Squares", "latex": "He taught that a square of one cell, in which unity was inserted, represented the unity and eternity of God; while the fact that a square of the second order could not be constructed illustrated the imperfection of the four elements, air, earth, fire, and water; and later writers added that it was symbolic of original sin.", "markdown": "He taught that a square of one cell, in which unity was inserted, represented the unity and eternity of God; while the fact that a square of the second order could not be constructed illustrated the imperfection of the four elements, air, earth, fire, and water; and later writers added that it was symbolic of original sin.", "why": "It shows how a mathematical fact (no second-order magic square exists) was given symbolic meaning, which suits a history note.", "use": [ "history", "website" ], "concepts": [ "concept/magic-square", "concept/order-of-a-magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-6b715607e6", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 162", "location": "Magic Squares", "latex": "The reason why such a square is magic can be explained best by expressing the numbers in the scale of notation whose radix is $5$ (or $n$, if the magic square is of the order $n$), except that $5$ is allowed to appear as a unit-digit and $0$ is not allowed to appear as a unit-digit.", "markdown": "The reason why such a square is magic can be explained best by expressing the numbers in the scale of notation whose radix is $5$ (or $n$, if the magic square is of the order $n$), except that $5$ is allowed to appear as a unit-digit and $0$ is not allowed to appear as a unit-digit.", "why": "Shows how a change of base explains why a construction gives equal sums, which is a strong idea for learners to see.", "use": [ "lesson" ], "concepts": [ "concept/denary-scale-of-notation", "concept/digit", "method/de-la-loub-re-s-method" ] }, { "id": "ball-mathematical-recreations-1905/x-79860dc816", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 164", "location": "Magic Squares", "latex": "The cells filled by the same number form a \\emph{broken diagonal}.", "markdown": "The cells filled by the same number form a *broken diagonal*.", "why": "Defines the broken diagonal in a short sentence that a learner can verify on the fifth-order example.", "use": [ "lesson" ], "concepts": [ "concept/broken-diagonal" ] }, { "id": "ball-mathematical-recreations-1905/x-e68a6a66e2", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 171", "location": "Magic Squares", "latex": "It is unfortunate that no more obvious rule---such, for instance, as one for bordering a doubly-even square---can be suggested for writing down instantly and without thought singly-even magic squares.", "markdown": "It is unfortunate that no more obvious rule---such, for instance, as one for bordering a doubly-even square---can be suggested for writing down instantly and without thought singly-even magic squares.", "why": "Shows the author openly admitting a gap in the theory, which models honest mathematical uncertainty for readers.", "use": [ "history", "website" ], "concepts": [ "concept/singly-even-magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-3d6ea74f75", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 159", "location": "Magic Squares", "latex": "The majority of the medieval astrologers and physicians were much impressed by such arrangements.", "markdown": "The majority of the medieval astrologers and physicians were much impressed by such arrangements.", "why": "Places magic squares in their historical setting, showing why they once carried mystical meaning.", "use": [ "history" ], "concepts": [ "concept/magic-square", "person/cornelius-agrippa" ] }, { "id": "ball-mathematical-recreations-1905/x-a379ac0fd8", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 172", "location": "Magic Squares", "latex": "The square so formed is necessarily magic in rows, columns, and diagonals.", "markdown": "The square so formed is necessarily magic in rows, columns, and diagonals.", "why": "It states the claim the whole construction exists to prove, so a learner knows what the two subsidiary squares are for.", "use": [ "lesson" ], "concepts": [ "concept/magic-square", "method/constructing-an-even-magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-c5ba3d5df6", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 171", "location": "Magic Squares", "latex": "two numbers which are equidistant from the ends of the series $1,2,3,\\dotsc,n$ are said to be \\emph{complementary}.", "markdown": "two numbers which are equidistant from the ends of the series $1,2,3,\\dotsc,n$ are said to be *complementary*.", "why": "It gives the precise meaning of complementary, the idea on which the whole even construction turns.", "use": [ "lesson" ], "concepts": [ "concept/complementary-numbers" ] }, { "id": "ball-mathematical-recreations-1905/x-385446ac83", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 173", "location": "Magic Squares", "latex": "In the case of a singly even square, that is, one in which $n$ is divisible by $2$, but not by $4$, we cannot satisfy the proviso if any horizontal row in the first square has all its vertically related squares, other than the two squares in the diagonals, filled with complementary numbers.", "markdown": "In the case of a singly even square, that is, one in which $n$ is divisible by $2$, but not by $4$, we cannot satisfy the proviso if any horizontal row in the first square has all its vertically related squares, other than the two squares in the diagonals, filled with complementary numbers.", "why": "It warns that the construction can fail for orders divisible by 2 but not by 4, a trap a learner would otherwise walk into.", "use": [ "lesson" ], "concepts": [ "concept/complementary-numbers", "concept/even-magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-5bf3779119", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 175", "location": "Magic Squares", "latex": "In this manner from the magic square of the $3$rd order we can build up successively squares of the orders $5$, $7$, $9$,~\\&c., that is, any odd magic square.", "markdown": "In this manner from the magic square of the $3$rd order we can build up successively squares of the orders $5$, $7$, $9$, &c., that is, any odd magic square.", "why": "It shows in one line how a bordering step lets a small magic square grow into larger ones of the same parity.", "use": [ "lesson", "website" ], "concepts": [ "concept/bordered-magic-square", "concept/magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-85743adf58", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 178", "location": "Magic Squares", "latex": "By reciprocating the figures composed of the points on which the numbers are placed we obtain a collection of lines forming pencils, and, if these lines be numbered to correspond with the points, the pencils will be magic", "markdown": "By reciprocating the figures composed of the points on which the numbers are placed we obtain a collection of lines forming pencils, and, if these lines be numbered to correspond with the points, the pencils will be magic", "why": "It gives a striking reversal: swapping points for lines turns a magic square into a magic arrangement of pencils.", "use": [ "website" ], "concepts": [ "concept/magic-pencil", "concept/pencil-of-lines" ] }, { "id": "ball-mathematical-recreations-1905/x-8cdf2e27d7", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 176", "location": "Magic Squares", "latex": "I believe that with a little patience a magic square of any order can be thus built up, and of course it will have the property that, if each border is successively stripped off, the square will still remain magic.", "markdown": "I believe that with a little patience a magic square of any order can be thus built up, and of course it will have the property that, if each border is successively stripped off, the square will still remain magic.", "why": "It is the author's own assurance that a bordered square keeps its magic property at every layer, which a learner can check on a small example.", "use": [ "history", "website" ], "concepts": [ "concept/bordered-magic-square" ] }, { "id": "ball-mathematical-recreations-1905/x-565585b221", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 199", "location": "Unicursal Problems", "latex": "in the determination of a route along the edges of a regular dodecahedron which will pass once and only once through every angular point.", "markdown": "in the determination of a route along the edges of a regular dodecahedron which will pass once and only once through every angular point.", "why": "It states the Hamiltonian game in one sentence a learner can hold: find a route along the edges that visits every corner exactly once.", "use": [ "lesson" ], "concepts": [ "concept/hamiltonian-cycle" ] }, { "id": "ball-mathematical-recreations-1905/x-f5a360f271", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 202", "location": "Unicursal Problems", "latex": "On a board containing an even number of cells the path may or may not be re-entrant, but on a board containing an odd number of cells it cannot be re-entrant.", "markdown": "On a board containing an even number of cells the path may or may not be re-entrant, but on a board containing an odd number of cells it cannot be re-entrant.", "why": "It gives a clear parity fact that learners can test on a small board before accepting it.", "use": [ "lesson" ], "concepts": [ "concept/knight-s-tour", "concept/re-entrant-route", "theorem/knight-s-move-alternates-colour" ] }, { "id": "ball-mathematical-recreations-1905/x-ba25f8db36", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 208", "location": "Unicursal Problems", "latex": "His rule is that the knight must be moved always to one of the cells from which it will command the fewest squares not already traversed.", "markdown": "His rule is that the knight must be moved always to one of the cells from which it will command the fewest squares not already traversed.", "why": "It gives a concrete greedy rule that a learner can apply move by move to build a knight's tour.", "use": [ "lesson" ], "concepts": [ "concept/knight-s-move", "method/warnsdorff-s-rule" ] }, { "id": "ball-mathematical-recreations-1905/x-26b151acfa", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 200", "location": "Unicursal Problems", "latex": "if we approach a point by one edge, the only routes open to us are one to the right, denoted by $r$, and one to the left, denoted by $l$.", "markdown": "if we approach a point by one edge, the only routes open to us are one to the right, denoted by $r$, and one to the left, denoted by $l$.", "why": "It shows how a route at a corner with three edges reduces to a choice between two symbols, which is the core of the symbolic method.", "use": [ "lesson" ], "concepts": [ "concept/hamiltonian-cycle", "concept/left-right-operation-notation" ] }, { "id": "ball-mathematical-recreations-1905/x-c7be2bfba8", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 208", "location": "Unicursal Problems", "latex": "The rule has not been proved to be true, but no exception to it is known", "markdown": "The rule has not been proved to be true, but no exception to it is known", "why": "It shows the reader that a widely used rule is an observed regularity, not a proved theorem, which teaches honest uncertainty.", "use": [ "lesson", "history" ], "concepts": [ "method/warnsdorff-s-rule" ] }, { "id": "ball-mathematical-recreations-1905/x-942f6faff4", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 201", "location": "Unicursal Problems", "latex": "It is convenient to make a mark or to put down a counter at each corner as soon as it is reached, and this will prevent our passing through the same town twice.", "markdown": "It is convenient to make a mark or to put down a counter at each corner as soon as it is reached, and this will prevent our passing through the same town twice.", "why": "It gives a practical habit that stops a learner from revisiting a vertex while tracing a route by hand.", "use": [ "lesson" ], "concepts": [ "concept/hamiltonian-cycle" ] }, { "id": "ball-mathematical-recreations-1905/x-59133abd89", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 207", "location": "Unicursal Problems", "latex": "Thus the cells $(x, y)$ and $(9-x, 9-y)$ are complementary, where $x$ and $y$ denote respectively the column and row occupied by the cell.", "markdown": "Thus the cells $(x, y)$ and $(9-x, 9-y)$ are complementary, where $x$ and $y$ denote respectively the column and row occupied by the cell.", "why": "It defines the complementary cell on an eight-by-eight board with explicit coordinates a learner can check.", "use": [ "lesson" ], "concepts": [ "concept/complementary-cell", "method/euler-s-method-for-the-knight-s-tour" ] }, { "id": "ball-mathematical-recreations-1905/x-0a01ac4c62", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 199", "location": "Unicursal Problems", "latex": "who invented this game---if game is the right term for it---denoted the twenty angular points on the solid by letters which stand for various towns.", "markdown": "who invented this game---if game is the right term for it---denoted the twenty angular points on the solid by letters which stand for various towns.", "why": "Its wry aside on whether a puzzle counts as a game gives a learner a memorable way into the history of the problem.", "use": [ "history", "website" ], "concepts": [ "concept/hamiltonian-cycle", "person/william-rowan-hamilton" ] }, { "id": "ball-mathematical-recreations-1905/x-2c63073b01", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 212", "location": "Unicursal Problems", "latex": "The annulus may be divided into four closed circuits, each containing $12$ cells: these are marked respectively with the letters $a$, $b$, $c$, $d$.", "markdown": "The annulus may be divided into four closed circuits, each containing $12$ cells: these are marked respectively with the letters $a$, $b$, $c$, $d$.", "why": "It shows how Moon's method cuts the outer ring into labelled circuits that can then be joined into one path.", "use": [ "lesson" ], "concepts": [ "concept/annulus", "concept/closed-circuit", "method/moon-s-method-for-the-knight-s-tour" ] }, { "id": "ball-mathematical-recreations-1905/x-efd4a32989", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 213", "location": "Unicursal Problems", "latex": "Thus if the initial cell is on $a$, we might take either of the cycles $a\\ D\\ b\\ C\\ d\\ A\\ c\\ B$, or $a\\ D\\ c\\ B\\ d\\ A\\ b\\ C$.", "markdown": "Thus if the initial cell is on $a$, we might take either of the cycles $a\\ D\\ b\\ C\\ d\\ A\\ c\\ B$, or $a\\ D\\ c\\ B\\ d\\ A\\ b\\ C$.", "why": "It gives a concrete sequence of circuit labels so a learner can follow how the joining rule works in practice.", "use": [ "lesson", "website" ], "concepts": [ "concept/closed-circuit", "method/moon-s-method-for-the-knight-s-tour" ] }, { "id": "ball-mathematical-recreations-1905/x-84714008de", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 213", "location": "Unicursal Problems", "latex": "By following these rules we always can connect the routes into one path, but in general it will not be re-entrant.", "markdown": "By following these rules we always can connect the routes into one path, but in general it will not be re-entrant.", "why": "It warns that a path built by Moon's joining rules may reach one path without returning to its start.", "use": [ "lesson" ], "concepts": [ "concept/re-entrant-route", "method/moon-s-method-for-the-knight-s-tour" ] }, { "id": "ball-mathematical-recreations-1905/x-0039e0cab3", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 213", "location": "Unicursal Problems", "latex": "It is convenient to take the cells in each circuit in one and the same direction, but a circuit in the outer annulus must not end in a corner cell, and to avoid this we may have to alter the direction in which a circuit is taken.", "markdown": "It is convenient to take the cells in each circuit in one and the same direction, but a circuit in the outer annulus must not end in a corner cell, and to avoid this we may have to alter the direction in which a circuit is taken.", "why": "It names a common pitfall, ending a circuit on a corner cell, and the fix of reversing direction.", "use": [ "lesson" ], "concepts": [ "concept/annulus", "concept/corner-cell", "method/moon-s-method-for-the-knight-s-tour" ] }, { "id": "ball-mathematical-recreations-1905/x-c8800b3640", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 214", "location": "Unicursal Problems", "latex": "and on the other hand is greater than $31,054144$---since this latter number is the number of re-entrant paths of a particular type", "markdown": "and on the other hand is greater than $31,054144$---since this latter number is the number of re-entrant paths of a particular type", "why": "It shows how the count of knight's paths is only bounded so far; the printed bound 31,054144 is quoted as transcribed and may carry a typesetting error, so verify it before use.", "use": [ "history" ], "concepts": [ "concept/number-of-knight-s-paths", "concept/re-entrant-route" ] }, { "id": "ball-mathematical-recreations-1905/x-dd49f1f5a8", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 213", "location": "Unicursal Problems", "latex": "It leads to eight forms, similar to that in the diagram printed \\vpageref{Jaenisch}, in which the sum of the numbers in every column and every row is $260$; but although symmetrical it is not in my opinion so easy to reproduce as that given by Roget.", "markdown": "It leads to eight forms, similar to that in the diagram printed Jaenisch, in which the sum of the numbers in every column and every row is $260$; but although symmetrical it is not in my opinion so easy to reproduce as that given by Roget.", "why": "It places Jaenisch's symmetrical magic-square solutions against Roget's method as a historical comparison of how easy each is to reproduce.", "use": [ "history", "website" ], "concepts": [ "concept/magic-square-cell", "method/jaenisch-s-method-for-the-knight-s-tour" ] }, { "id": "ball-mathematical-recreations-1905/x-dc328b26d5", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 213", "location": "Unicursal Problems", "latex": "It is as yet impossible to say how many solutions of the problem exist.", "markdown": "It is as yet impossible to say how many solutions of the problem exist.", "why": "It tells learners that the total number of knight's tours was an open question in the book's time.", "use": [ "history" ], "concepts": [ "concept/knight-s-tour", "concept/number-of-knight-s-paths" ] }, { "id": "ball-mathematical-recreations-1905/x-e64763a29b", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 185", "location": "Unicursal Problems", "latex": "It is evident that the question will not be affected if we suppose the islands to diminish to points and the bridges to lengthen out. In this way we ultimately obtain a geometrical figure or network.", "markdown": "It is evident that the question will not be affected if we suppose the islands to diminish to points and the bridges to lengthen out. In this way we ultimately obtain a geometrical figure or network.", "why": "It shows the step that turns a town's bridges into a network of points and lines, which is the idea the whole chapter rests on.", "use": [ "lesson" ], "concepts": [ "concept/edge", "concept/k-nigsberg-bridge-problem", "concept/node", "concept/unicursal-figure" ] }, { "id": "ball-mathematical-recreations-1905/x-a64c2d32a7", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 187", "location": "Unicursal Problems", "latex": "Since a node of the $n$th order is one at which $n$ branches meet, there are $n$ hooks there. Also since the figure is closed, $n$ cannot be less than $2$.", "markdown": "Since a node of the $n$th order is one at which $n$ branches meet, there are $n$ hooks there. Also since the figure is closed, $n$ cannot be less than $2$.", "why": "It gives the counting argument behind the parity theorem, so a learner can see why the order of a node matters.", "use": [ "lesson" ], "concepts": [ "concept/closed-network", "concept/order-of-a-node", "theorem/odd-nodes-of-a-closed-network-are-even-in-number" ] }, { "id": "ball-mathematical-recreations-1905/x-75911dbe1b", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 187", "location": "Unicursal Problems", "latex": "The number of hooks at each node is even, and if they are unfastened they can be re-coupled together in pairs, the arrangement of the pairs being immaterial.", "markdown": "The number of hooks at each node is even, and if they are unfastened they can be re-coupled together in pairs, the arrangement of the pairs being immaterial.", "why": "It states the key observation that makes a figure with no odd node traversable, phrased with physical hooks and string.", "use": [ "lesson", "website" ], "concepts": [ "concept/even-node", "method/re-coupling-of-hooks", "theorem/unicursal-route-for-a-figure-with-no-odd-node" ] }, { "id": "ball-mathematical-recreations-1905/x-f0839f57c8", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 198", "location": "Unicursal Problems", "latex": "The number of trees with $n$ given nodes is $n^{n-2}$.", "markdown": "The number of trees with $n$ given nodes is $n^{n-2}$.", "why": "It is a compact closed-form counting result that shows how trees can be counted without drawing them.", "use": [ "lesson" ], "concepts": [ "concept/tree" ] }, { "id": "ball-mathematical-recreations-1905/x-eed573a30e", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 192", "location": "Unicursal Problems", "latex": "Of course to walk twice over every path in a labyrinth is not the shortest way of arriving at the centre, but, if it is performed correctly, the whole maze is traversed, the arrival at the centre at some point in the course of the route is certain, and it is impossible to lose one's way", "markdown": "Of course to walk twice over every path in a labyrinth is not the shortest way of arriving at the centre, but, if it is performed correctly, the whole maze is traversed, the arrival at the centre at some point in the course of the route is certain, and it is impossible to lose one’s way", "why": "It is a vivid, honest contrast between the shortest route and a guaranteed one, which suits a general audience.", "use": [ "website" ], "concepts": [ "concept/maze", "method/maze-traversal-rule" ] }, { "id": "ball-mathematical-recreations-1905/x-cca5528aa6", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 190", "location": "Unicursal Problems", "latex": "said to have been originally traced in the sand by the point of his scimetar without taking the scimetar off the ground or retracing any part of the figure", "markdown": "said to have been originally traced in the sand by the point of his scimetar without taking the scimetar off the ground or retracing any part of the figure", "why": "The tale of a figure drawn without lifting the pen gives a memorable historical hook for why only-even-node figures are traversable.", "use": [ "history", "website" ], "concepts": [ "concept/even-node", "concept/unicursal-figure" ] }, { "id": "ball-mathematical-recreations-1905/x-5490b84afd", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 191", "location": "Unicursal Problems", "latex": "a chess-board, divided as usual by straight lines into $64$ cells, has $28$ odd nodes and $53$ even nodes: hence it would require $14$ separate pen-strokes to trace out all the boundaries without going over any more than once.", "markdown": "a chess-board, divided as usual by straight lines into $64$ cells, has $28$ odd nodes and $53$ even nodes: hence it would require $14$ separate pen-strokes to trace out all the boundaries without going over any more than once.", "why": "It applies the counting rule to a familiar object and shows the consequence in pen-strokes, which makes the theorem concrete.", "use": [ "lesson", "website" ], "concepts": [ "concept/even-node", "concept/odd-node", "theorem/2n-odd-nodes-need-n-routes" ] }, { "id": "ball-mathematical-recreations-1905/x-e7715cc976", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 215", "location": "The Mathematical Tripos", "latex": "``No man of science should think it a waste of time to learn something of the history of his own subject; nor is the investigation of laborious methods now fallen into disuse, or of errors once commonly accepted the least valuable of mental disciplines.''", "markdown": "“No man of science should think it a waste of time to learn something of the history of his own subject; nor is the investigation of laborious methods now fallen into disuse, or of errors once commonly accepted the least valuable of mental disciplines.”", "why": "A learner is invited to see the history of a subject, and its old errors, as part of learning it.", "use": [ "history", "website" ], "concepts": [ "concept/mathematics" ] }, { "id": "ball-mathematical-recreations-1905/x-c1750611a8", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 217", "location": "The Mathematical Tripos", "latex": "Probably the science (as distinct from the art) of mathematics, save so far as involved in the quadrivium, was still an exotic study, but it was not wholly neglected.", "markdown": "Probably the science (as distinct from the art) of mathematics, save so far as involved in the quadrivium, was still an exotic study, but it was not wholly neglected.", "why": "It separates mathematics as a science from the older practical art, which helps a reader see how the subject was once understood.", "use": [ "history" ], "concepts": [ "concept/mathematics" ] }, { "id": "ball-mathematical-recreations-1905/x-e9b557869b", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 217", "location": "The Mathematical Tripos", "latex": "Newton's\\index{Newton} mathematical career dates from 1665; his reputation, abilities, and influence attracted general attention to the subject.", "markdown": "Newton’s mathematical career dates from 1665; his reputation, abilities, and influence attracted general attention to the subject.", "why": "It dates Newton's mathematical work and shows how one figure drew wide attention to the subject.", "use": [ "history" ], "concepts": [ "person/isaac-newton" ] }, { "id": "ball-mathematical-recreations-1905/x-7f3ee8ce06", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 227", "location": "The Mathematical Tripos", "latex": "The extraction of roots, the arithmetic of surds, the invention of divisers, the resolution of quadratic, cubic, and biquadratic equations;", "markdown": "The extraction of roots, the arithmetic of surds, the invention of divisers, the resolution of quadratic, cubic, and biquadratic equations;", "why": "It gives a dated list of the algebra and arithmetic a Cambridge candidate was expected to know in 1772.", "use": [ "history" ], "concepts": [ "concept/arithmetic", "concept/cubic-equation", "concept/quadratic-equation" ] }, { "id": "ball-mathematical-recreations-1905/x-fe4332394e", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 219", "location": "The Mathematical Tripos", "latex": "I was badgered for two hours with arguments given and answered in Latin,---or what we call Latin---against Newton's first section, Lagrange's derived functions, and Locke on innate principles.", "markdown": "I was badgered for two hours with arguments given and answered in Latin,---or what we call Latin---against Newton’s first section, Lagrange’s derived functions, and Locke on innate principles.", "why": "De Morgan's account conveys how demanding the old disputations were, a vivid picture for general readers.", "use": [ "history", "website" ], "concepts": [] }, { "id": "ball-mathematical-recreations-1905/x-e37dc8df0a", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 228", "location": "The Mathematical Tripos", "latex": "and according to tradition, on one occasion the candidates on entering in the morning found the ink in the pots on their desks frozen.", "markdown": "and according to tradition, on one occasion the candidates on entering in the morning found the ink in the pots on their desks frozen.", "why": "A charming detail that shows the physical hardship of the old examination.", "use": [ "website", "history" ], "concepts": [] }, { "id": "ball-mathematical-recreations-1905/x-d7c72baa1f", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 230", "location": "The Mathematical Tripos", "latex": "In the cubic equation $x^3 + qx + r = 0$ which wants the second term;\nsupposing $x = a + b$ and $3ab=-q$, to determine the value of $x$.", "markdown": "In the cubic equation $x^3 + qx + r = 0$ which wants the second term; supposing $x = a + b$ and $3ab=-q$, to determine the value of $x$.", "why": "A concrete 1786 Senate-House problem that shows a learner how a cubic can be attacked by an assumed split into two parts.", "use": [ "lesson" ], "concepts": [ "concept/cubic-equation", "concept/equation" ] }, { "id": "ball-mathematical-recreations-1905/x-ce0ba96fc9", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 237", "location": "The Mathematical Tripos", "latex": "I was not up to the \\emph{differential}\ncalculus,\nand never acquired it.", "markdown": "I was not up to the *differential* calculus, and never acquired it.", "why": "Pollock's candid remark separates the fluxional method of his day from the differential calculus, which helps a reader place the two notations historically.", "use": [ "history" ], "concepts": [ "concept/calculus", "concept/fluxional-notation" ] }, { "id": "ball-mathematical-recreations-1905/x-11ec03ee3a", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 233", "location": "The Mathematical Tripos", "latex": "they would require every candidate to show a competent\nknowledge of the first book of Euclid, arithmetic, vulgar and\ndecimal fractions, simple and quadratic equations,", "markdown": "they would require every candidate to show a competent knowledge of the first book of Euclid, arithmetic, vulgar and decimal fractions, simple and quadratic equations,", "why": "The 1799 pass standard lists exactly what a candidate was expected to know, giving a clear picture of the entry level of the period.", "use": [ "history", "website" ], "concepts": [ "concept/arithmetic", "concept/decimal-fraction", "concept/quadratic-equation", "concept/rational-number", "person/euclid" ] }, { "id": "ball-mathematical-recreations-1905/x-f9d40bcd5f", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 239", "location": "The Mathematical Tripos", "latex": "`It was a quadratic,' said he, `made\nme senior wrangler.'", "markdown": "‘It was a quadratic,’ said he, ‘made me senior wrangler.’", "why": "Turner's remark is a memorable reminder that a quadratic equation could decide the top place in the examination.", "use": [ "website", "history" ], "concepts": [ "concept/quadratic-equation" ] }, { "id": "ball-mathematical-recreations-1905/x-a0cb48a10b", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 241", "location": "The Mathematical Tripos", "latex": "They created an Analytical Society which Babbage explained was formed to advocate ``the principles of pure $d$-ism as opposed to the \\emph{dot}-age of the University.''", "markdown": "They created an Analytical Society which Babbage explained was formed to advocate “the principles of pure $d$-ism as opposed to the *dot*-age of the University.”", "why": "A vivid contemporary slogan that shows the choice of differential notation over Newton's dots was a deliberate reform.", "use": [ "history", "website" ], "concepts": [ "concept/analytical-society", "concept/fluxional-notation", "concept/mathematical-notation", "person/charles-babbage" ] }, { "id": "ball-mathematical-recreations-1905/x-91acd04955", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 241", "location": "The Mathematical Tripos", "latex": "In 1817 Peacock, who was moderator, introduced the symbols for differentiation into the papers set in the Senate-House examination.", "markdown": "In 1817 Peacock, who was moderator, introduced the symbols for differentiation into the papers set in the Senate-House examination.", "why": "It dates the first appearance of differential symbols in the Cambridge examination papers.", "use": [ "history", "lesson" ], "concepts": [ "concept/mathematical-notation", "concept/senate-house-examination", "method/differentiation", "person/george-peacock" ] }, { "id": "ball-mathematical-recreations-1905/x-25c04b851f", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 242", "location": "The Mathematical Tripos", "latex": "It is by silent perseverance only that we can hope to reduce the many-headed monster of prejudice, and make the University answer her character as the loving mother of good learning and science.", "markdown": "It is by silent perseverance only that we can hope to reduce the many-headed monster of prejudice, and make the University answer her character as the loving mother of good learning and science.", "why": "Peacock's letter shows how reformers of the period framed curriculum change as a long campaign.", "use": [ "history", "website" ], "concepts": [ "concept/mathematical-tripos", "person/george-peacock" ] }, { "id": "ball-mathematical-recreations-1905/x-1e6fc1d27b", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 243", "location": "The Mathematical Tripos", "latex": "We are employed from seven in the morning till five in the evening in giving out questions and receiving written answers to them; and when that is over, we have to read over all the papers which we have received in the course of the day, to determine who have done best, which is a business that in numerous years has often kept the examiners up the half of every night; but this year is not particularly numerous.", "markdown": "We are employed from seven in the morning till five in the evening in giving out questions and receiving written answers to them; and when that is over, we have to read over all the papers which we have received in the course of the day, to determine who have done best, which is a business that in numerous years has often kept the examiners up the half of every night; but this year is not particularly numerous.", "why": "Whewell's description gives a concrete picture of how a nineteenth-century written examination was run and marked.", "use": [ "history", "website" ], "concepts": [ "concept/mathematical-tripos", "person/william-whewell" ] }, { "id": "ball-mathematical-recreations-1905/x-e8f924ac14", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 250", "location": "The Mathematical Tripos", "latex": "The assignment of marks to groups of subjects was made under the impression that the best candidates would concentrate their abilities on a selection of subjects from the various divisions. But it was found that, unless the questions were made extremely difficult, more marks could be obtained by reading superficially all the subjects in the five divisions than by attaining real proficiency in a few of the higher ones:", "markdown": "The assignment of marks to groups of subjects was made under the impression that the best candidates would concentrate their abilities on a selection of subjects from the various divisions. But it was found that, unless the questions were made extremely difficult, more marks could be obtained by reading superficially all the subjects in the five divisions than by attaining real proficiency in a few of the higher ones:", "why": "It is a cautionary case of a scoring rule rewarding breadth over depth, which teachers can use when discussing assessment design.", "use": [ "lesson" ], "concepts": [ "concept/mathematical-tripos" ] }, { "id": "ball-mathematical-recreations-1905/x-8960e69a4c", "chapter": "ball-mathematical-recreations-1905/ch-vii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 254", "location": "The Mathematical Tripos", "latex": "In 1895 the proctors and moderators, without consulting the Senate, sent in no verses, and thus, in spite of widespread regret, an interesting custom of many centuries standing was destroyed.", "markdown": "In 1895 the proctors and moderators, without consulting the Senate, sent in no verses, and thus, in spite of widespread regret, an interesting custom of many centuries standing was destroyed.", "why": "It shows how a long-standing tradition can end quietly when no one is formally responsible for it.", "use": [ "history" ], "concepts": [ "concept/mathematical-tripos" ] }, { "id": "ball-mathematical-recreations-1905/x-0f652319a3", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 255", "location": "Three Geometrical Problems", "latex": "\\textsc{Among} the more interesting geometrical problems of antiquity are three questions which attracted the special attention of the early Greek mathematicians.", "markdown": "Among the more interesting geometrical problems of antiquity are three questions which attracted the special attention of the early Greek mathematicians.", "why": "It introduces the three classical problems and why they have lasted in the history of geometry.", "use": [ "lesson", "history" ], "concepts": [ "concept/duplication-of-the-cube", "concept/squaring-the-circle", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/x-bd1ad02bd8", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 255", "location": "Three Geometrical Problems", "latex": "To duplicate a cube the length of whose side is $a$, we have to find a line of length $x$, such that $x^3 = 2a^3$.", "markdown": "To duplicate a cube the length of whose side is $a$, we have to find a line of length $x$, such that $x^3 = 2a^3$.", "why": "It turns the duplication problem into a single equation a learner can check.", "use": [ "lesson" ], "concepts": [ "concept/cube", "concept/duplication-of-the-cube" ] }, { "id": "ball-mathematical-recreations-1905/x-03b43af4ff", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 258", "location": "Three Geometrical Problems", "latex": "He did not give a geometrical construction, but he reduced the question to that of finding two means between one straight line ($a$), and another twice as long ($2a$). If these means are $x$ and $y$, we have $a: x = x: y = y: 2a$, from which it follows that $x^3 = 2a^3$. It is in this form that the problem is always presented now.", "markdown": "He did not give a geometrical construction, but he reduced the question to that of finding two means between one straight line ($a$), and another twice as long ($2a$). If these means are $x$ and $y$, we have $a: x = x: y = y: 2a$, from which it follows that $x^3 = 2a^3$. It is in this form that the problem is always presented now.", "why": "It shows how a geometric problem becomes a proportion, which is the form the problem still takes today.", "use": [ "lesson", "history" ], "concepts": [ "concept/duplication-of-the-cube", "concept/means-of-a-proportion" ] }, { "id": "ball-mathematical-recreations-1905/x-b14842756c", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 257", "location": "Three Geometrical Problems", "latex": "The insertion of Plato's name is an obvious anachronism.", "markdown": "The insertion of Plato’s name is an obvious anachronism.", "why": "It warns learners to test a legend against its dates before trusting it.", "use": [ "history" ], "concepts": [ "concept/duplication-of-the-cube", "person/plato" ] }, { "id": "ball-mathematical-recreations-1905/x-2b2598d4f4", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 257", "location": "Three Geometrical Problems", "latex": "It is probable that the Greeks were aware that the latter ratio is incommensurable, in other words, that no two integers can be found whose ratio is the same as that of $\\sqrt[3]{2}: 1$, but it did not therefore follow that they could not find the ratio by geometry: in fact, the side and diagonal of a square are instances of lines whose numerical measures are incommensurable.", "markdown": "It is probable that the Greeks were aware that the latter ratio is incommensurable, in other words, that no two integers can be found whose ratio is the same as that of $\\sqrt[3]{2}: 1$, but it did not therefore follow that they could not find the ratio by geometry: in fact, the side and diagonal of a square are instances of lines whose numerical measures are incommensurable.", "why": "It corrects the common mistake that an incommensurable ratio cannot be constructed with a ruler and compass.", "use": [ "lesson" ], "concepts": [ "concept/duplication-of-the-cube", "concept/incommensurable-magnitudes" ] }, { "id": "ball-mathematical-recreations-1905/x-290c978456", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 266", "location": "Three Geometrical Problems", "latex": "It is however a mere accident that $\\pi$ is defined usually in that way, and it really represents a certain number which would enter into analysis from whatever side the subject was approached.", "markdown": "It is however a mere accident that $\\pi$ is defined usually in that way, and it really represents a certain number which would enter into analysis from whatever side the subject was approached.", "why": "It tells learners that pi is a fundamental number and not only a ratio of a circle's circumference to its diameter.", "use": [ "lesson", "website" ], "concepts": [ "concept/pi" ] }, { "id": "ball-mathematical-recreations-1905/x-63445412de", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 267", "location": "Three Geometrical Problems", "latex": "In reality the fact that the ratio of the length of the circumference of a circle to its diameter is the number denoted by $\\pi$ does not afford the best analytical definition of $\\pi$, and is only one of its properties.", "markdown": "In reality the fact that the ratio of the length of the circumference of a circle to its diameter is the number denoted by $\\pi$ does not afford the best analytical definition of $\\pi$, and is only one of its properties.", "why": "It shows that a familiar definition is only one property of a number, which encourages deeper study.", "use": [ "lesson", "website" ], "concepts": [ "concept/pi" ] }, { "id": "ball-mathematical-recreations-1905/x-f2f7bacebd", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 267", "location": "Three Geometrical Problems", "latex": "The use of a single symbol to denote this number $3.14159\\ldots$ seems to have been introduced about the beginning of the eighteenth century.", "markdown": "The use of a single symbol to denote this number $3.14159\\ldots$ seems to have been introduced about the beginning of the eighteenth century.", "why": "It dates the modern symbol for pi and gives the learner a sense of how notation grew over time.", "use": [ "history" ], "concepts": [ "concept/pi" ] }, { "id": "ball-mathematical-recreations-1905/x-166dc7e372", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 267", "location": "Three Geometrical Problems", "latex": "We may say that the $\\pi$-calculators who used the first method regarded $\\pi$ as equivalent to a geometrical ratio, but those who adopted the modern method treated it as the symbol for a certain number which enters into numerous branches of mathematical analysis.", "markdown": "We may say that the $\\pi$-calculators who used the first method regarded $\\pi$ as equivalent to a geometrical ratio, but those who adopted the modern method treated it as the symbol for a certain number which enters into numerous branches of mathematical analysis.", "why": "It contrasts the geometric and analytic views of pi so a learner sees why the number matters beyond circles.", "use": [ "lesson" ], "concepts": [ "concept/pi", "method/polygon-perimeter-method" ] }, { "id": "ball-mathematical-recreations-1905/x-64182d0beb", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 269", "location": "Three Geometrical Problems", "latex": "With a polygon of $n$ sides this process gives a value of $\\pi$ correct to at least the integral part of $(2\\log n - 1.19)$ places of decimals.", "markdown": "With a polygon of $n$ sides this process gives a value of $\\pi$ correct to at least the integral part of $(2\\log n - 1.19)$ places of decimals.", "why": "It shows how the number of polygon sides controls the precision of an approximation, a clear worked link between method and accuracy.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "concept/integer-part", "concept/pi", "concept/regular-polygon" ] }, { "id": "ball-mathematical-recreations-1905/x-eb68e7624e", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 273", "location": "Three Geometrical Problems", "latex": "The reason is that Archimedes, having calculated the lengths of the sides of inscribed and circumscribed regular polygons of $n$ sides, assumed that the length of $1/n$th of the perimeter of the circle was intermediate between them; whereas Snell constructed from the sides of these polygons two other lines which gave closer limits for the corresponding arc.", "markdown": "The reason is that Archimedes, having calculated the lengths of the sides of inscribed and circumscribed regular polygons of $n$ sides, assumed that the length of $1/n$th of the perimeter of the circle was intermediate between them; whereas Snell constructed from the sides of these polygons two other lines which gave closer limits for the corresponding arc.", "why": "It explains why one method of bounding a circle's arc beats another, teaching the idea of closer limits.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "concept/pi", "concept/regular-polygon" ] }, { "id": "ball-mathematical-recreations-1905/x-c9cbe33140", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 277", "location": "Three Geometrical Problems", "latex": "If the experiment is repeated many hundreds of times, the ratio of the number of favourable cases to the whole number of experiments will be very nearly equal to this fraction: hence the value of $\\pi$ can be found.", "markdown": "If the experiment is repeated many hundreds of times, the ratio of the number of favourable cases to the whole number of experiments will be very nearly equal to this fraction: hence the value of $\\pi$ can be found.", "why": "It shows a surprising route to pi through chance, letting a learner connect probability with a fixed constant.", "use": [ "lesson" ], "concepts": [ "concept/pi", "concept/probability" ] }, { "id": "ball-mathematical-recreations-1905/x-4bac922b2a", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 277", "location": "Three Geometrical Problems", "latex": "Inscribe in the given circle a square, and to three times the diameter of the circle add a fifth of a side of the square, the result will differ from the circumference of the circle by less than one-seventeen-thousandth part of it.", "markdown": "Inscribe in the given circle a square, and to three times the diameter of the circle add a fifth of a side of the square, the result will differ from the circumference of the circle by less than one-seventeen-thousandth part of it.", "why": "It gives a simple geometric construction that approximates the circumference to a tight tolerance, inviting the learner to test it.", "use": [ "website" ], "concepts": [ "concept/approximation", "concept/circle", "concept/pi", "quantity/circumference" ] }, { "id": "ball-mathematical-recreations-1905/x-24569f54ec", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 277", "location": "Three Geometrical Problems", "latex": "``Only prove to me that it is impossible,'' said one of them, ``and I will set about it immediately''; and doubtless the statement that the problem is insoluble has attracted much attention to it.", "markdown": "“Only prove to me that it is impossible,” said one of them, “and I will set about it immediately”; and doubtless the statement that the problem is insoluble has attracted much attention to it.", "why": "It is a vivid old remark about squaring the circle that brings out the human side of the problem's history.", "use": [ "history" ], "concepts": [ "concept/pi" ] }, { "id": "ball-mathematical-recreations-1905/x-871ceb45ae", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 280", "location": "Mersenne's Numbers", "latex": "It is evident that, if $p$ is not a prime, then $N$ is composite, and two or more of its factors can be written down by inspection.", "markdown": "It is evident that, if $p$ is not a prime, then $N$ is composite, and two or more of its factors can be written down by inspection.", "why": "It shows a learner that a composite exponent already yields factors of 2^p-1 with no search at all.", "use": [ "lesson" ], "concepts": [ "concept/factor", "concept/mersenne-number", "concept/prime-number", "method/writing-by-inspection" ] }, { "id": "ball-mathematical-recreations-1905/x-286c137789", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 292", "location": "Mersenne's Numbers", "latex": "We know by Fermat's Theorem that if $x + 1$ is a prime then $2^x-1$ is divisible by $x + 1$.", "markdown": "We know by Fermat’s Theorem that if $x + 1$ is a prime then $2^x-1$ is divisible by $x + 1$.", "why": "It gives the one-line fact that drives the chapter's seventh method for finding divisors.", "use": [ "lesson" ], "concepts": [ "theorem/fermat-s-little-theorem" ] }, { "id": "ball-mathematical-recreations-1905/x-f471bbedfe", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 292", "location": "Mersenne's Numbers", "latex": "The number $N$ when expressed in the binary scale, consists of $1$ repeated $p$ times.", "markdown": "The number $N$ when expressed in the binary scale, consists of $1$ repeated $p$ times.", "why": "It makes the link between 2^p-1 and base 2 visible, so a learner sees why the binary scale was proposed.", "use": [ "lesson", "website" ], "concepts": [ "concept/binary-scale", "concept/mersenne-number" ] }, { "id": "ball-mathematical-recreations-1905/x-3394fa6d4d", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 283", "location": "Mersenne's Numbers", "latex": "that, if $4n + 3$ and $8n + 7$ are primes, then $2^{4n+3}-1 \\equiv 0 \\pmod{8n + 7}$.", "markdown": "that, if $4n + 3$ and $8n + 7$ are primes, then $2^{4n+3}-1 \\equiv 0 \\pmod{8n + 7}$.", "why": "It states the one general theorem on the subject, in the form Euler enunciated and Lagrange proved.", "use": [ "history", "lesson" ], "concepts": [ "concept/congruence", "theorem/euler-s-theorem-on-factors-of-mersenne-numbers" ] }, { "id": "ball-mathematical-recreations-1905/x-96f0c80f17", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 283", "location": "Mersenne's Numbers", "latex": "Comme de $25$, qui est un quarré, ôtez $2$; le reste $23$ mesurera la 11\\textsuperscript{e} puissance $-1$;", "markdown": "Comme de $25$, qui est un quarré, ôtez $2$; le reste $23$ mesurera la 11e puissance $-1$;", "why": "It is Fermat's own 1640 wording of the rule that produces the factors of 2^11-1, a direct window onto the history.", "use": [ "history", "website" ], "concepts": [ "concept/mersenne-number", "person/pierre-de-fermat" ] }, { "id": "ball-mathematical-recreations-1905/x-c6fd53b2ee", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 286", "location": "Mersenne's Numbers", "latex": "but the riddle as to how it was discovered is still, after nearly 250 years, unsolved.", "markdown": "but the riddle as to how it was discovered is still, after nearly 250 years, unsolved.", "why": "It is an honest admission that a famous result's origin is unknown, which shows learners that open questions exist.", "use": [ "history", "website" ], "concepts": [ "person/marin-mersenne" ] }, { "id": "ball-mathematical-recreations-1905/x-e9f313a3a4", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 280", "location": "Mersenne's Numbers", "latex": "Qui vndecim alios repererit, nouerit se analysim omnem, quae fuerit hactenus, superasse:", "markdown": "Qui vndecim alios repererit, nouerit se analysim omnem, quae fuerit hactenus, superasse:", "why": "It is Mersenne's own boast about the difficulty of the subject, a charming turn of phrase for a history page.", "use": [ "history", "website" ], "concepts": [ "person/marin-mersenne" ] }, { "id": "ball-mathematical-recreations-1905/x-1045ef21b2", "chapter": "ball-mathematical-recreations-1905/ch-x", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 297", "location": "Astrology", "latex": "This sphere was divided into twelve equal spaces by great circles drawn through the zenith, the angle between any two consecutive circles being $30^{\\circ}$.", "markdown": "This sphere was divided into twelve equal spaces by great circles drawn through the zenith, the angle between any two consecutive circles being $30^{\\circ}$.", "why": "It shows how a sphere is divided into equal spaces by great circles, a geometric construction the learner can picture and check.", "use": [ "lesson" ], "concepts": [ "concept/great-circle", "concept/sphere" ] }, { "id": "ball-mathematical-recreations-1905/x-41e9151b50", "chapter": "ball-mathematical-recreations-1905/ch-x", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 302", "location": "Astrology", "latex": "It must be remembered also that the rules were laid down at a time when men were unacquainted with any exact science, with the possible exception of mathematics, and further that, if astrology had been reduced to a series of inelastic rules applicable to all horoscopes, the number of failures to predict the future correctly would have rapidly led to a recognition of the folly of the art.", "markdown": "It must be remembered also that the rules were laid down at a time when men were unacquainted with any exact science, with the possible exception of mathematics, and further that, if astrology had been reduced to a series of inelastic rules applicable to all horoscopes, the number of failures to predict the future correctly would have rapidly led to a recognition of the folly of the art.", "why": "It gives a clear argument that a rule which is too precise would be falsified quickly, which is a useful point about testing any claimed theory against outcomes.", "use": [ "lesson", "history" ], "concepts": [ "concept/astrology", "concept/mathematics" ] }, { "id": "ball-mathematical-recreations-1905/x-41398b5e33", "chapter": "ball-mathematical-recreations-1905/ch-x", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 302", "location": "Astrology", "latex": "This is the less necessary as the rules---especially as to the relative importance to be assigned to various planets when their influence was conflicting---were so vague that astrologers had little difficulty in finding in the horoscope of a client any fact about his life of which they had information or any trait of character which they suspected him to possess.", "markdown": "This is the less necessary as the rules---especially as to the relative importance to be assigned to various planets when their influence was conflicting---were so vague that astrologers had little difficulty in finding in the horoscope of a client any fact about his life of which they had information or any trait of character which they suspected him to possess.", "why": "It shows how vague rules let a reader find confirmation in any outcome, a warning about unfalsifiable predictions that learners can apply to their own reasoning.", "use": [ "lesson", "website" ], "concepts": [ "concept/astrology", "concept/horoscope" ] }, { "id": "ball-mathematical-recreations-1905/x-d773587e74", "chapter": "ball-mathematical-recreations-1905/ch-x", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 306", "location": "Astrology", "latex": "It is easier to give instances of success in horoscopy than of failure. Not only are all ambiguous predictions esteemed to be successful, but it is notorious that prophecies which have been verified by the subsequent course of events are remembered and quoted, while the far more numerous instances in which the prophecies have been falsified are forgotten or passed over in silence.", "markdown": "It is easier to give instances of success in horoscopy than of failure. Not only are all ambiguous predictions esteemed to be successful, but it is notorious that prophecies which have been verified by the subsequent course of events are remembered and quoted, while the far more numerous instances in which the prophecies have been falsified are forgotten or passed over in silence.", "why": "It names the selection bias that makes hits look more common than misses, a point worth teaching when judging any evidence.", "use": [ "lesson", "website" ], "concepts": [ "concept/astrology" ] }, { "id": "ball-mathematical-recreations-1905/x-837f078ce6", "chapter": "ball-mathematical-recreations-1905/ch-x", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 305", "location": "Astrology", "latex": "The old lady dug in the spot thus indicated, and found her property; and it may be conjectured that she believed in astrology for the rest of her life.", "markdown": "The old lady dug in the spot thus indicated, and found her property; and it may be conjectured that she believed in astrology for the rest of her life.", "why": "It closes the Flamsteed story with a vivid example of how a lucky result can persuade a believer, which is good material for a history reading.", "use": [ "history", "website" ], "concepts": [ "concept/astrology", "person/john-flamsteed" ] }, { "id": "ball-mathematical-recreations-1905/x-06997623fc", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 311", "location": "Cryptographs and Ciphers", "latex": "The art of constructing a cryptograph lies in the concealment of the proper order of the essential letters or words: the art of constructing a cipher lies in concealing what letters or words are represented by the symbols used.", "markdown": "The art of constructing a cryptograph lies in the concealment of the proper order of the essential letters or words: the art of constructing a cipher lies in concealing what letters or words are represented by the symbols used.", "why": "It states plainly the difference between the two methods: a cryptograph hides the order of the letters, a cipher hides which letter each symbol stands for.", "use": [ "lesson" ], "concepts": [ "concept/cipher", "concept/cryptograph" ] }, { "id": "ball-mathematical-recreations-1905/x-80a4555193", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 311", "location": "Cryptographs and Ciphers", "latex": "A simple example is when each letter is replaced by the one that immediately follows it in the natural order of the alphabet, \\emph{a} being replaced by \\emph{b}, \\emph{b} by \\emph{c}, and so on, and finally \\emph{z} by \\emph{a}.", "markdown": "A simple example is when each letter is replaced by the one that immediately follows it in the natural order of the alphabet, *a* being replaced by *b*, *b* by *c*, and so on, and finally *z* by *a*.", "why": "A concrete shift cipher a learner can try by hand before meeting the general idea of a key.", "use": [ "lesson" ], "concepts": [ "concept/cipher", "concept/key" ] }, { "id": "ball-mathematical-recreations-1905/x-c644cb21bb", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 312", "location": "Cryptographs and Ciphers", "latex": "The majority of stories dealing with secret communications are concerned with the artfulness with which the message is concealed or conveyed and have nothing to do with cryptographs or ciphers.", "markdown": "The majority of stories dealing with secret communications are concerned with the artfulness with which the message is concealed or conveyed and have nothing to do with cryptographs or ciphers.", "why": "It separates hiding a message from encoding it, which is the test the chapter uses throughout.", "use": [ "lesson", "website" ], "concepts": [ "concept/cipher", "concept/cryptograph", "concept/cryptography" ] }, { "id": "ball-mathematical-recreations-1905/x-2bb1ab2b9b", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 318", "location": "Cryptographs and Ciphers", "latex": "The defect of the method is that the broken letters at once attract attention and suggest the system used.", "markdown": "The defect of the method is that the broken letters at once attract attention and suggest the system used.", "why": "A clear warning that a visible oddity in a message can give away the method used to hide it.", "use": [ "lesson" ], "concepts": [ "concept/cryptograph", "instrument/scytale" ] }, { "id": "ball-mathematical-recreations-1905/x-b63b92df0a", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 320", "location": "Cryptographs and Ciphers", "latex": "It is said that during the Indian Mutiny messages in English, but written in Greek characters, were used freely, and successfully baffled the ingenuity of the enemy, into whose hands they fell.", "markdown": "It is said that during the Indian Mutiny messages in English, but written in Greek characters, were used freely, and successfully baffled the ingenuity of the enemy, into whose hands they fell.", "why": "A historical anecdote showing a cipher built by changing only the alphabet, with the reported success stated as hearsay.", "use": [ "history" ], "concepts": [ "concept/cipher" ] }, { "id": "ball-mathematical-recreations-1905/x-313814a71f", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 322", "location": "Cryptographs and Ciphers", "latex": "In English the letter which occurs most frequently is \\emph{e}.", "markdown": "In English the letter which occurs most frequently is *e*.", "why": "It opens the letter-frequency clues that let a learner begin breaking a substitution cipher.", "use": [ "lesson" ], "concepts": [ "method/frequency-analysis" ] }, { "id": "ball-mathematical-recreations-1905/x-21f10aa65b", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 324", "location": "Cryptographs and Ciphers", "latex": "A cipher of the second type is one in which the same letter or word is, in some or all cases, represented by more than one symbol, and this symbol always represents the same letter or word.", "markdown": "A cipher of the second type is one in which the same letter or word is, in some or all cases, represented by more than one symbol, and this symbol always represents the same letter or word.", "why": "It gives the learner the clean definition that separates a homophonic cipher from the others.", "use": [ "lesson" ], "concepts": [ "concept/cipher", "concept/homophonic-cipher" ] }, { "id": "ball-mathematical-recreations-1905/x-f180ed1910", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 324", "location": "Cryptographs and Ciphers", "latex": "A disadvantage of this cipher is that since each letter is denoted by two symbols the length of the message is doubled by putting it in cipher.", "markdown": "A disadvantage of this cipher is that since each letter is denoted by two symbols the length of the message is doubled by putting it in cipher.", "why": "It shows the cost of a cipher in message length, a trade-off a learner can check by counting symbols.", "use": [ "lesson" ], "concepts": [ "concept/cipher", "concept/digit" ] }, { "id": "ball-mathematical-recreations-1905/x-25464e8aa9", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 328", "location": "Cryptographs and Ciphers", "latex": "He took a key-word, such as \\emph{prudentia}, and constructed as many alphabets as there were letters in it, each alphabet being arranged cyclically and beginning respectively with the letters $p$, $r$, $u$, $d$, $e$, $n$, $t$, $i$, and $a$.", "markdown": "He took a key-word, such as *prudentia*, and constructed as many alphabets as there were letters in it, each alphabet being arranged cyclically and beginning respectively with the letters $p$, $r$, $u$, $d$, $e$, $n$, $t$, $i$, and $a$.", "why": "It describes how a key-word fixes a set of cyclic alphabets, which a learner can rebuild by hand.", "use": [ "lesson", "website" ], "concepts": [ "concept/alphabet", "concept/key", "concept/polyalphabetic-cipher" ] }, { "id": "ball-mathematical-recreations-1905/x-43aa59e6f1", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 325", "location": "Cryptographs and Ciphers", "latex": "cells, and each cell is determined uniquely by the two letters denoting its row and column.", "markdown": "cells, and each cell is determined uniquely by the two letters denoting its row and column.", "why": "It states the table idea in one line: a pair of key letters fixes one cell, and so one symbol.", "use": [ "lesson" ], "concepts": [ "concept/cipher", "concept/key" ] }, { "id": "ball-mathematical-recreations-1905/x-4def7fbc08", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 331", "location": "Cryptographs and Ciphers", "latex": "Lastly, no ambiguity should be possible in deciphering the communication.", "markdown": "Lastly, no ambiguity should be possible in deciphering the communication.", "why": "It names the one requirement that rules out the fourth type, giving the reason for the classification.", "use": [ "lesson" ], "concepts": [ "concept/cipher-requisites", "concept/many-to-one-cipher" ] }, { "id": "ball-mathematical-recreations-1905/x-41616d7ca8", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 335", "location": "Cryptographs and Ciphers", "latex": "For twenty-four hours de~Rohan pored over the message, but, failing to read it, he admitted his guilt, and was executed November~27, 1674.", "markdown": "For twenty-four hours de Rohan pored over the message, but, failing to read it, he admitted his guilt, and was executed November 27, 1674.", "why": "A short cipher with few clues can defeat its reader, which illustrates why the key matters.", "use": [ "history" ], "concepts": [ "concept/cipher", "concept/key" ] }, { "id": "ball-mathematical-recreations-1905/x-07018d380d", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 333", "location": "Cryptographs and Ciphers", "latex": "The Queen seems to have found writing in cipher a great trouble.", "markdown": "The Queen seems to have found writing in cipher a great trouble.", "why": "A charming remark about a historical correspondent that makes the cipher's human cost vivid.", "use": [ "history", "website" ], "concepts": [ "concept/cipher", "person/charles-i" ] }, { "id": "ball-mathematical-recreations-1905/x-c8bbf37465", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 337", "location": "Cryptographs and Ciphers", "latex": "A code dictionary is prepared in which every word likely to be used is printed, and the words are numbered consecutively $00000$, $00001, \\ldots$ up, if necessary, to $99999$. Thus each word is represented by a number of five digits, and there are $10^5$ such numbers available.", "markdown": "A code dictionary is prepared in which every word likely to be used is printed, and the words are numbered consecutively $00000$, $00001, \\ldots$ up, if necessary, to $99999$. Thus each word is represented by a number of five digits, and there are $10^5$ such numbers available.", "why": "It gives the learner the whole mechanism of a book cipher in two sentences: a dictionary, consecutive numbering, and five-digit codes.", "use": [ "lesson" ], "concepts": [ "concept/book-cipher", "concept/digit" ] }, { "id": "ball-mathematical-recreations-1905/x-6557bd1602", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 337", "location": "Cryptographs and Ciphers", "latex": "This is a cipher with $10^5$ symbols, and as each symbol consists of five digits, a message of $n$ words is denoted by $5n$ digits, and probably is not longer than the message when written in the ordinary way.", "markdown": "This is a cipher with $10^5$ symbols, and as each symbol consists of five digits, a message of $n$ words is denoted by $5n$ digits, and probably is not longer than the message when written in the ordinary way.", "why": "It shows the learner how the size of the symbol set sets the length of the coded message.", "use": [ "lesson" ], "concepts": [ "concept/book-cipher", "concept/digit" ] }, { "id": "ball-mathematical-recreations-1905/x-66ec4f7a86", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 338", "location": "Cryptographs and Ciphers", "latex": "If the clue number is the same all through the message it would be possible by not more than $10^5$ trials to discover the message.", "markdown": "If the clue number is the same all through the message it would be possible by not more than $10^5$ trials to discover the message.", "why": "It warns the learner that a fixed key is weak, because an attacker can simply try every possibility.", "use": [ "lesson" ], "concepts": [ "concept/book-cipher", "concept/key" ] }, { "id": "ball-mathematical-recreations-1905/x-fce91791af", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 337", "location": "Cryptographs and Ciphers", "latex": "When a code message is published by the Government receiving it, the construction of the sentences is usually altered before publication, so that the key may not be discoverable by anyone in possession of the code-book or who has seen the cipher message.", "markdown": "When a code message is published by the Government receiving it, the construction of the sentences is usually altered before publication, so that the key may not be discoverable by anyone in possession of the code-book or who has seen the cipher message.", "why": "It explains why a published message must be reworded, so that the key cannot be recovered from it.", "use": [ "lesson", "history" ], "concepts": [ "concept/book-cipher", "concept/key" ] }, { "id": "ball-mathematical-recreations-1905/x-2c1f83c06e", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 338", "location": "Cryptographs and Ciphers", "latex": "For if we make a similar code with the twenty-six letters of the alphabet instead of the ten digits, four letters for each word or phrase would give us $26^4$, that is, $456976$ possible variations.", "markdown": "For if we make a similar code with the twenty-six letters of the alphabet instead of the ten digits, four letters for each word or phrase would give us $26^4$, that is, $456976$ possible variations.", "why": "It shows how changing the symbol set changes the number of possible code words, using a simple power calculation.", "use": [ "lesson" ], "concepts": [ "concept/alphabet", "concept/book-cipher", "concept/digit" ] }, { "id": "ball-mathematical-recreations-1905/x-49ec722d57", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 339", "location": "Cryptographs and Ciphers", "latex": "Poe\\index{Poe, E.A.} wrote an essay on cryptography in which he said that it may be roundly asserted that human ingenuity cannot concoct a cipher which human ingenuity cannot resolve---a conclusion which is hardly justified by the known facts.", "markdown": "Poe wrote an essay on cryptography in which he said that it may be roundly asserted that human ingenuity cannot concoct a cipher which human ingenuity cannot resolve---a conclusion which is hardly justified by the known facts.", "why": "It gives a memorable historical claim about ciphers and invites the learner to test it against the known facts.", "use": [ "history", "website" ], "concepts": [ "concept/cipher", "concept/cryptography", "person/edgar-allan-poe" ] }, { "id": "ball-mathematical-recreations-1905/x-928984a2f8", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 339", "location": "Cryptographs and Ciphers", "latex": "More than one of his correspondents did not play the game fairly, not only employing foreign languages, but using several different ciphers in the same communication.", "markdown": "More than one of his correspondents did not play the game fairly, not only employing foreign languages, but using several different ciphers in the same communication.", "why": "It shows the historical practice of testing ciphers by sending them to a public challenge, and how correspondents tried to defeat the solver.", "use": [ "history" ], "concepts": [ "concept/cipher", "person/edgar-allan-poe" ] }, { "id": "ball-mathematical-recreations-1905/x-265817c877", "chapter": "ball-mathematical-recreations-1905/ch-xii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 341", "location": "Hyper-space", "latex": "Space, time, and matter cannot be defined; but the means of measuring them and the investigation of their properties fall within the domain of mathematics.", "markdown": "Space, time, and matter cannot be defined; but the means of measuring them and the investigation of their properties fall within the domain of mathematics.", "why": "It states plainly what a mathematician can and cannot claim about space, time and matter, which is a useful opening for a lesson on what geometry is about.", "use": [ "lesson" ], "concepts": [ "concept/mathematics" ] }, { "id": "ball-mathematical-recreations-1905/x-f105beeec5", "chapter": "ball-mathematical-recreations-1905/ch-xii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 350", "location": "Hyper-space", "latex": "In the hyperbolic system there are no similar figures of unequal size; the area of a triangle can be deduced from the sum of its angles, which is always less than two right angles; and there is a finite maximum to the area of a triangle.", "markdown": "In the hyperbolic system there are no similar figures of unequal size; the area of a triangle can be deduced from the sum of its angles, which is always less than two right angles; and there is a finite maximum to the area of a triangle.", "why": "It gives three concrete consequences of hyperbolic geometry that a learner can check against the familiar Euclidean triangle.", "use": [ "lesson", "website" ], "concepts": [ "concept/hyperbolic-geometry", "theorem/area-of-a-triangle", "theorem/sum-of-the-angles-of-a-triangle" ] }, { "id": "ball-mathematical-recreations-1905/x-9e8c4c8356", "chapter": "ball-mathematical-recreations-1905/ch-xii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 342", "location": "Hyper-space", "latex": "If an inhabitant of flatland was able to move in three dimensions, he would be credited with supernatural powers by those who were unable so to move; for he could appear or disappear at will, could (so far as they could tell) create matter or destroy it, and would be free from so many constraints to which the other inhabitants were subject that his actions would be inexplicable by them.", "markdown": "If an inhabitant of flatland was able to move in three dimensions, he would be credited with supernatural powers by those who were unable so to move; for he could appear or disappear at will, could (so far as they could tell) create matter or destroy it, and would be free from so many constraints to which the other inhabitants were subject that his actions would be inexplicable by them.", "why": "The flatland analogy makes the idea of an extra dimension vivid by showing how a higher-dimensional move would look like magic to a lower-dimensional observer.", "use": [ "website", "history" ], "concepts": [ "concept/dimension", "concept/physical-analogy" ] }, { "id": "ball-mathematical-recreations-1905/x-943ec7c623", "chapter": "ball-mathematical-recreations-1905/ch-xii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 348", "location": "Hyper-space", "latex": "I have assumed that the thickness of the supporting space is small and uniform, because then the intensity of the energy transmitted from a source to any point would vary inversely as the square of the distance, as is the case; whereas if the supporting space was a body of four dimensions, the law would be that of the inverse cube of the distance.", "markdown": "I have assumed that the thickness of the supporting space is small and uniform, because then the intensity of the energy transmitted from a source to any point would vary inversely as the square of the distance, as is the case; whereas if the supporting space was a body of four dimensions, the law would be that of the inverse cube of the distance.", "why": "It shows a physical consequence that depends on the number of dimensions, so a learner sees how dimension changes an inverse-square law.", "use": [ "lesson" ], "concepts": [ "concept/dimension", "concept/intensity" ] }, { "id": "ball-mathematical-recreations-1905/x-b4c9fdd4cd", "chapter": "ball-mathematical-recreations-1905/ch-xii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 349", "location": "Hyper-space", "latex": "which is usually stated in the form that if a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, then these straight lines being continually produced will at length meet upon that side on which are the angles which are less than two right angles.", "markdown": "which is usually stated in the form that if a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, then these straight lines being continually produced will at length meet upon that side on which are the angles which are less than two right angles.", "why": "It gives the exact form of Euclid's parallel axiom that all later non-Euclidean geometries modify, so a learner has the statement to be varied.", "use": [ "lesson" ], "concepts": [ "concept/parallel-lines", "theorem/parallel-postulate" ] }, { "id": "ball-mathematical-recreations-1905/x-7dfe7c1887", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 353", "location": "Time and its Measurement", "latex": "Thus to measure a length we may take a foot-rule, and by applying it to the given length as often as is necessary, we shall find how many feet the length contains.", "markdown": "Thus to measure a length we may take a foot-rule, and by applying it to the given length as often as is necessary, we shall find how many feet the length contains.", "why": "It shows the basic idea of measurement, a repeated unit compared against the quantity, using a familiar ruler.", "use": [ "lesson" ], "concepts": [ "instrument/foot-rule", "quantity/length" ] }, { "id": "ball-mathematical-recreations-1905/x-6541445974", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 356", "location": "Time and its Measurement", "latex": "The true solar day is not however always of the same duration.", "markdown": "The true solar day is not however always of the same duration.", "why": "It explains why a day measured by the sun must be averaged before it can serve as a clock unit.", "use": [ "lesson" ], "concepts": [ "concept/mean-solar-day", "concept/true-solar-day" ] }, { "id": "ball-mathematical-recreations-1905/x-9df2cff115", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 365", "location": "Time and its Measurement", "latex": "I believe it is not generally known that a sun-dial can be so constructed that the shadow will, for a short time near sunrise and sunset, move backwards on the dial", "markdown": "I believe it is not generally known that a sun-dial can be so constructed that the shadow will, for a short time near sunrise and sunset, move backwards on the dial", "why": "It is a surprising fact that invites a learner to reason about shadows and the sun's apparent path.", "use": [ "website" ], "concepts": [ "instrument/sun-dial" ] }, { "id": "ball-mathematical-recreations-1905/x-996a122b30", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 359", "location": "Time and its Measurement", "latex": "The Julian calendar made the year, on an average, contain 365.25 days. The actual value is, very approximately, 365.242216 days.", "markdown": "The Julian calendar made the year, on an average, contain 365.25 days. The actual value is, very approximately, 365.242216 days.", "why": "It gives a concrete numerical error that a learner can check against the difference between two averages.", "use": [ "lesson" ], "concepts": [ "concept/julian-calendar" ] }, { "id": "ball-mathematical-recreations-1905/x-c1685ae55f", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 357", "location": "Time and its Measurement", "latex": "The French were the last civilized nation to abandon the use of true time: this was in 1816.", "markdown": "The French were the last civilized nation to abandon the use of true time: this was in 1816.", "why": "It gives a dated historical fact about the adoption of clock time.", "use": [ "history" ], "concepts": [ "concept/mean-solar-day" ] }, { "id": "ball-mathematical-recreations-1905/x-76a56a21c6", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 355", "location": "Time and its Measurement", "latex": "Our experiences are consistent with this statement, and that is as high an authority as a mathematician hopes to get.", "markdown": "Our experiences are consistent with this statement, and that is as high an authority as a mathematician hopes to get.", "why": "It shows how a unit of time is justified by physical reasoning and experience rather than by direct proof.", "use": [ "lesson", "history" ], "concepts": [ "law/inertia", "quantity/time" ] }, { "id": "ball-mathematical-recreations-1905/x-811e3172b3", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 368", "location": "Time and its Measurement", "latex": "The time occupied by a given amount of some liquid or sand in running through a given orifice under the same conditions is always the same, and by noting the level of the liquid which has run through the orifice, or which remains to run through it, a measure of time can be obtained.", "markdown": "The time occupied by a given amount of some liquid or sand in running through a given orifice under the same conditions is always the same, and by noting the level of the liquid which has run through the orifice, or which remains to run through it, a measure of time can be obtained.", "why": "It shows a learner, without any apparatus beyond a container, how a steady flow turns into a measure of time.", "use": [ "lesson" ], "concepts": [ "concept/measure-of-time", "instrument/hourglass", "instrument/water-clock" ] }, { "id": "ball-mathematical-recreations-1905/x-c283ac394a", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 369", "location": "Time and its Measurement", "latex": "Most of these early clocks were regulated by horizontal balances: pendulums being then unknown.", "markdown": "Most of these early clocks were regulated by horizontal balances: pendulums being then unknown.", "why": "It gives a dated fact that places the pendulum's arrival in clock history, useful for a lesson on how a technology develops.", "use": [ "history" ], "concepts": [ "concept/pendulum", "instrument/clock" ] }, { "id": "ball-mathematical-recreations-1905/x-dc76ba32f7", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 371", "location": "Time and its Measurement", "latex": "so arranged that the cylinder descended with uniform velocity between two vertical pillars on which the hours were marked at equidistant intervals.", "markdown": "so arranged that the cylinder descended with uniform velocity between two vertical pillars on which the hours were marked at equidistant intervals.", "why": "It presents uniform motion as a practical device, letting a learner see constant velocity as a timekeeping principle.", "use": [ "lesson", "website" ], "concepts": [ "concept/uniform-motion", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/x-25f2bb7297", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 372", "location": "Time and its Measurement", "latex": "Both move round in the same direction, but the angular velocity of the hour-hand is twice as great as that of the sun. Hence the rule.", "markdown": "Both move round in the same direction, but the angular velocity of the hour-hand is twice as great as that of the sun. Hence the rule.", "why": "It gives the reason for the watch-compass rule in one comparison of angular velocities, which a learner can check with a turning clock.", "use": [ "lesson", "website" ], "concepts": [ "method/finding-south-from-a-watch", "quantity/angular-velocity" ] }, { "id": "ball-mathematical-recreations-1905/x-7baa90162f", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 372", "location": "Time and its Measurement", "latex": "To do this it is sufficient to point the hour-hand to the sun, and then the direction which bisects the angle between the hour and the figure \\textsc{xii} will point due south.", "markdown": "To do this it is sufficient to point the hour-hand to the sun, and then the direction which bisects the angle between the hour and the figure xii will point due south.", "why": "It states the watch-compass method as a procedure a learner can carry out outdoors.", "use": [ "lesson", "website" ], "concepts": [ "concept/plane-angle", "method/finding-south-from-a-watch" ] }, { "id": "ball-mathematical-recreations-1905/x-de0c78fa19", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 380", "location": "Matter and Ether Theories", "latex": "If an observer lived in two dimensional space filled with ether and confined by two parallel and adjacent surfaces, and if through a hole in one of these surfaces fresh ether were squirted into this space, the variations of pressure thereby produced might give the impression of a hard impenetrable body.", "markdown": "If an observer lived in two dimensional space filled with ether and confined by two parallel and adjacent surfaces, and if through a hole in one of these surfaces fresh ether were squirted into this space, the variations of pressure thereby produced might give the impression of a hard impenetrable body.", "why": "A concrete two-dimensional thought experiment that lets a learner see how a pressure variation could feel like a solid object.", "use": [ "lesson", "website" ], "concepts": [ "concept/atom", "concept/ether", "quantity/pressure" ] }, { "id": "ball-mathematical-recreations-1905/x-590fd0d345", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 378", "location": "Matter and Ether Theories", "latex": "The tendency of the particles forming a ring to maintain their annular connection may be illustrated by placing such a box on one side of a room in a direct line with the flame of a lighted candle on the other side.", "markdown": "The tendency of the particles forming a ring to maintain their annular connection may be illustrated by placing such a box on one side of a room in a direct line with the flame of a lighted candle on the other side.", "why": "It describes a home experiment with smoke rings that makes the idea of a vortex ring tangible.", "use": [ "lesson" ], "concepts": [ "concept/vortex-ring" ] }, { "id": "ball-mathematical-recreations-1905/x-5f7ad95733", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 387", "location": "Matter and Ether Theories", "latex": "A body by itself in space would receive on an average as many blows on one side as on another, and therefore would have no tendency to move.", "markdown": "A body by itself in space would receive on an average as many blows on one side as on another, and therefore would have no tendency to move.", "why": "It opens Le Sage's shadowing argument by showing why a lone body feels no net push.", "use": [ "lesson" ], "concepts": [ "law/law-of-gravity" ] }, { "id": "ball-mathematical-recreations-1905/x-78c815093c", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 376", "location": "Matter and Ether Theories", "latex": "It is essential to the theory that the atom shall have a mass but shall not have dimensions.", "markdown": "It is essential to the theory that the atom shall have a mass but shall not have dimensions.", "why": "It states plainly the strange requirement of Boscovich's atom, which helps learners see how far it departs from the popular picture.", "use": [ "lesson" ], "concepts": [ "concept/atom", "quantity/mass" ] }, { "id": "ball-mathematical-recreations-1905/x-1c3e7a5cc1", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 378", "location": "Matter and Ether Theories", "latex": "This is true as far as it goes, but it is not more unreasonable than to attribute all the fundamental properties of matter to the atoms themselves, as is done by many writers.", "markdown": "This is true as far as it goes, but it is not more unreasonable than to attribute all the fundamental properties of matter to the atoms themselves, as is done by many writers.", "why": "It warns learners against judging dynamical atom theories as the only ones that assume something hard to believe.", "use": [ "lesson" ], "concepts": [ "concept/atom", "concept/hypothesis" ] }, { "id": "ball-mathematical-recreations-1905/x-de1ce17d62", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 379", "location": "Matter and Ether Theories", "latex": "Sir William Thomson gave the number of vibrations per second of a sodium ring as probably being greater than $10^{14}$.", "markdown": "Sir William Thomson gave the number of vibrations per second of a sodium ring as probably being greater than $10^{14}$.", "why": "It gives a historical sense of the enormous frequencies that were proposed for atomic vibration.", "use": [ "history" ], "concepts": [ "concept/vortex-ring", "person/william-thomson" ] }, { "id": "ball-mathematical-recreations-1905/x-37e84257ba", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 389", "location": "Matter and Ether Theories", "latex": "I should sum up the effect of this discussion on gravity on the relative probabilities of the hypotheses as to the constitution of matter enumerated above, by saying that it does not enable us to discriminate between them.", "markdown": "I should sum up the effect of this discussion on gravity on the relative probabilities of the hypotheses as to the constitution of matter enumerated above, by saying that it does not enable us to discriminate between them.", "why": "It models honest reporting of an open question, showing that a hypothesis set can remain undecided.", "use": [ "lesson", "history" ], "concepts": [ "concept/hypothesis", "law/law-of-gravity" ] }, { "id": "ball-mathematical-recreations-1905/x-4fc721b312", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 391", "location": "Matter and Ether Theories", "latex": "Thus, if the disturbance is represented by the swinging of a pendulum in a resisting medium, it might be supposed that the atoms were formed at the points of maximum amplitude, and we should expect that the atoms successively thrown off would form a series having the properties of its successive members connected by a regular periodic law.", "markdown": "Thus, if the disturbance is represented by the swinging of a pendulum in a resisting medium, it might be supposed that the atoms were formed at the points of maximum amplitude, and we should expect that the atoms successively thrown off would form a series having the properties of its successive members connected by a regular periodic law.", "why": "It shows a learner how a physical model (a swinging pendulum) is used to frame a testable conjecture about periodic structure.", "use": [ "lesson" ], "concepts": [ "concept/maximum", "concept/pendulum", "concept/periodic-function", "concept/phase" ] }, { "id": "ball-mathematical-recreations-1905/x-9dc52ebaa7", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 392", "location": "Matter and Ether Theories", "latex": "I should say rather that we have obtained an analogy which is sufficiently like the truth to suggest new discoveries. Such analogies are often the precursors of laws, so that it is not unreasonable to hope that ere long our knowledge of this border-land of chemistry and physics may be more definite, and thus that molecular physics may be brought within the domain of mathematics.", "markdown": "I should say rather that we have obtained an analogy which is sufficiently like the truth to suggest new discoveries. Such analogies are often the precursors of laws, so that it is not unreasonable to hope that ere long our knowledge of this border-land of chemistry and physics may be more definite, and thus that molecular physics may be brought within the domain of mathematics.", "why": "It teaches the difference between a useful analogy and a verified theory, a point a learner can take into any model-building.", "use": [ "lesson", "history" ], "concepts": [ "concept/physical-analogy" ] }, { "id": "ball-mathematical-recreations-1905/x-3f563735a2", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 390", "location": "Matter and Ether Theories", "latex": "just as two clocks whose rates are nearly the same tend to go at the same rate if their cases are connected.", "markdown": "just as two clocks whose rates are nearly the same tend to go at the same rate if their cases are connected.", "why": "A concrete picture of coupled oscillators that makes the idea of physical connection between molecules easy to grasp.", "use": [ "lesson" ], "concepts": [ "concept/matter", "concept/physical-analogy" ] }, { "id": "ball-mathematical-recreations-1905/x-877a98ab61", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 390", "location": "Matter and Ether Theories", "latex": "The objection to this is that no explanation is offered as to what has become of the excluded molecules.", "markdown": "The objection to this is that no explanation is offered as to what has become of the excluded molecules.", "why": "It models how a hypothesis is criticised: a rival explanation must account for what it discards.", "use": [ "lesson" ], "concepts": [ "concept/infinite-set", "concept/molecule" ] }, { "id": "ball-mathematical-recreations-1905/x-8f885cac54", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 394", "location": "Matter and Ether Theories", "latex": "Thus it would seem that a cubic inch of gas at ordinary pressure and temperature contains about $3 \\times 10^{20}$ molecules, all similar and equal, and each molecule has a volume of about $1/(3 \\times 10^{25})$th of a cubic inch;", "markdown": "Thus it would seem that a cubic inch of gas at ordinary pressure and temperature contains about $3 \\times 10^{20}$ molecules, all similar and equal, and each molecule has a volume of about $1/(3 \\times 10^{25})$th of a cubic inch;", "why": "It gives a vivid worked estimate of molecular number and volume from kinetic theory that learners can check by arithmetic.", "use": [ "lesson", "website" ], "concepts": [ "concept/kinetic-theory-of-heat", "concept/molecule", "quantity/number-of-molecules", "quantity/volume" ] }, { "id": "ball-mathematical-recreations-1905/x-4fadfb6d2b", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 394", "location": "Matter and Ether Theories", "latex": "and probably the size of a molecule would be about the same as that of a fives-ball.", "markdown": "and probably the size of a molecule would be about the same as that of a fives-ball.", "why": "A homely comparison that makes an unimaginably small size tangible, in the old style of Victorian popular science.", "use": [ "website", "history" ], "concepts": [ "concept/approximation", "concept/molecule" ] } ], "equations": [ { "id": "ball-mathematical-recreations-1905/eq-c63f8278d7", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 24", "location": "Some Arithmetical Questions", "latex": "A = M(a') + 1", "name": null, "statement": "A is a multiple of the product of the other moduli and exceeds a multiple of a' by one.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A", "meaning": "number chosen to be a multiple of b'c'd'... and to exceed a multiple of a' by unity" }, { "unit": null, "symbol": "a'", "meaning": "one of the numbers a', b', c', ... that are prime to one another" }, { "unit": null, "symbol": "M(x)", "meaning": "a multiple of x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/multiple", "concept/prime-to-one-another", "concept/remainder" ] }, { "id": "ball-mathematical-recreations-1905/eq-ec6bf95e11", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 24", "location": "Some Arithmetical Questions", "latex": "Aa = M(a') + a", "name": null, "statement": "The product Aa leaves remainder a when divided by a'.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "number that is a multiple of b'c'd'... and exceeds a multiple of a' by unity" }, { "unit": null, "symbol": "a", "meaning": "remainder when the chosen number n is divided by a'" }, { "unit": null, "symbol": "M(x)", "meaning": "a multiple of x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/divisibility", "concept/multiple", "concept/remainder" ] }, { "id": "ball-mathematical-recreations-1905/eq-687aeb9c19", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 24", "location": "Some Arithmetical Questions", "latex": "N = Aa + Bb + Cc + \\dotsb", "name": null, "statement": "N is the sum of each remainder multiplied by its chosen coefficient.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "N", "meaning": "the sum Aa + Bb + Cc + ..." }, { "unit": null, "symbol": "A, B, C", "meaning": "numbers found so that each is a multiple of the product of the other moduli and exceeds a multiple of its own modulus by unity" }, { "unit": null, "symbol": "a, b, c", "meaning": "remainders when n is divided by a', b', c', ..." } ], "sympy": "Eq(N, A*a + B*b + C*c)", "physics": false, "states": [], "concepts": [ "concept/multiple", "concept/remainder", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-fe0b12eb8d", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 24", "location": "Some Arithmetical Questions", "latex": "N-n &= M(a')\\,.", "name": null, "statement": "N and the chosen number n differ by a multiple of a'.", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "the sum Aa + Bb + Cc + ..." }, { "unit": null, "symbol": "n", "meaning": "the number selected, less than p" }, { "unit": null, "symbol": "M(x)", "meaning": "a multiple of x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/difference", "concept/divisibility", "concept/multiple" ] }, { "id": "ball-mathematical-recreations-1905/eq-cb79e0d70c", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 24", "location": "Some Arithmetical Questions", "latex": "N &= M(p) + n\\,.", "name": null, "statement": "N equals a multiple of p plus the chosen number n, so N leaves remainder n on division by p.", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "the sum Aa + Bb + Cc + ..." }, { "unit": null, "symbol": "p", "meaning": "product a'b'c'... of the moduli" }, { "unit": null, "symbol": "n", "meaning": "the number selected, less than p" }, { "unit": null, "symbol": "M(x)", "meaning": "a multiple of x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/multiple", "concept/product", "concept/remainder" ] }, { "id": "ball-mathematical-recreations-1905/eq-771c5e1c88", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 23", "location": "Some Arithmetical Questions", "latex": "x = e", "name": null, "statement": "The hundreds digit of the quotient Q is the tens digit x of the selected number.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "tens digit of the selected number 9x + y" }, { "unit": null, "symbol": "e", "meaning": "third digit from the right of the quotient obtained in the last division" } ], "sympy": "Eq(x, e)", "physics": false, "states": [], "concepts": [ "concept/denary-scale-of-notation", "concept/digit" ] }, { "id": "ball-mathematical-recreations-1905/eq-3b62ba80df", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 23", "location": "Some Arithmetical Questions", "latex": "y=9-r", "name": null, "statement": "The units digit y of the selected number equals nine minus the remainder r.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "units digit of the selected number 9x + y" }, { "unit": null, "symbol": "r", "meaning": "remainder when a-b+3(c-d) is divided by 9" } ], "sympy": "Eq(y, 9 - r)", "physics": false, "states": [], "concepts": [ "concept/denary-scale-of-notation", "concept/digit", "concept/remainder" ] }, { "id": "ball-mathematical-recreations-1905/eq-f86c8063cd", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 23", "location": "Some Arithmetical Questions", "latex": "9m-y = a-b + 3(c-d)", "name": null, "statement": "Nine times the integer m minus the units digit y equals a-b plus three times (c-d).", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "quotient obtained in the final division step" }, { "unit": null, "symbol": "y", "meaning": "units digit of the selected number" }, { "unit": null, "symbol": "a, b, c, d", "meaning": "the remainders and small chosen numbers in the fourth method" } ], "sympy": "Eq(9*m - y, a - b + 3*(c - d))", "physics": false, "states": [], "concepts": [ "concept/difference", "concept/digit", "concept/remainder" ] }, { "id": "ball-mathematical-recreations-1905/eq-633ff585e8", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 34", "location": "Some Arithmetical Questions", "latex": "(13 - x_1) + (13 - x_2) + \\dotsb + (13 - x_p) + r &= 52", "name": null, "statement": "The cards in the p piles (each pile has 13 - x_i cards) plus the r cards left over make up the whole pack of 52 cards.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_i", "meaning": "number of pips on the bottom card of pile i" }, { "unit": null, "symbol": "p", "meaning": "number of piles formed" }, { "unit": null, "symbol": "r", "meaning": "number of cards left over" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/real-number", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-011a01715c", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 34", "location": "Some Arithmetical Questions", "latex": "x_1 + x_2 + \\dotsb + x_p &= 13p - 52 + r", "name": null, "statement": "The total number of pips on the bottom cards of the p piles equals 13(p - 4) + r.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_i", "meaning": "number of pips on the bottom card of pile i" }, { "unit": null, "symbol": "p", "meaning": "number of piles formed" }, { "unit": null, "symbol": "r", "meaning": "number of cards left over" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/remainder", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-9ef4d4b4e7", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 42", "location": "Some Arithmetical Questions", "latex": "\\log(1 + x) = x - \\tfrac{1}{2}x^2 + \\tfrac{1}{3}x^3 - \\dotsb", "name": null, "statement": "The logarithm of 1 + x is written as the infinite series x - x^2/2 + x^3/3 - ...", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable of the series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/infinite-sequence", "concept/logarithm" ] }, { "id": "ball-mathematical-recreations-1905/eq-401be09736", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 42", "location": "Some Arithmetical Questions", "latex": "\\log 1 = 0", "name": null, "statement": "The logarithm of 1 is zero.", "kind": "result", "symbols": [], "sympy": "Eq(log(1), 0)", "physics": false, "states": [], "concepts": [ "concept/logarithm", "theorem/logarithm-of-1-is-zero" ] }, { "id": "ball-mathematical-recreations-1905/eq-e94a4ace65", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 44", "location": "Some Arithmetical Questions", "latex": "\\sqrt{x-y} = i \\sqrt{y-x}", "name": null, "statement": "As an identity in x and y, the square root of x - y equals i times the square root of y - x, where i is either +sqrt(-1) or -sqrt(-1).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a number (any value, as an identity holds for all)" }, { "unit": null, "symbol": "y", "meaning": "a number (any value, as an identity holds for all)" }, { "unit": null, "symbol": "i", "meaning": "either +sqrt(-1) or -sqrt(-1)" } ], "sympy": "Eq(sqrt(x - y), i*sqrt(y - x))", "physics": false, "states": [], "concepts": [ "concept/identity", "concept/real-number", "concept/square" ] }, { "id": "ball-mathematical-recreations-1905/eq-dff140c395", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 42", "location": "Some Arithmetical Questions", "latex": "a + b &= 2c", "name": "arithmetic mean", "statement": "If c is the arithmetic mean of a and b, then a + b equals 2c.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "one of two unequal numbers" }, { "unit": null, "symbol": "b", "meaning": "the other of two unequal numbers" }, { "unit": null, "symbol": "c", "meaning": "arithmetic mean of a and b" } ], "sympy": "Eq(a + b, 2*c)", "physics": false, "states": [ "concept/arithmetical-mean" ], "concepts": [ "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-ca50a6b923", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 44", "location": "Some Arithmetical Questions", "latex": "a:b = c:d", "name": null, "statement": "The ratio a to b is the same as the ratio c to d, so a, b, c, d are in proportion.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "first term of the proportion" }, { "unit": null, "symbol": "b", "meaning": "second term of the proportion" }, { "unit": null, "symbol": "c", "meaning": "third term of the proportion" }, { "unit": null, "symbol": "d", "meaning": "fourth term of the proportion" } ], "sympy": "Eq(a/b, c/d)", "physics": false, "states": [], "concepts": [ "concept/proportion" ] }, { "id": "ball-mathematical-recreations-1905/eq-f91984f04f", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 44", "location": "Some Arithmetical Questions", "latex": "ad=bc", "name": null, "statement": "If the product of two numbers equals the product of two others, the four numbers are in proportion.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "first number of the first pair" }, { "unit": null, "symbol": "d", "meaning": "second number of the first pair" }, { "unit": null, "symbol": "b", "meaning": "first number of the second pair" }, { "unit": null, "symbol": "c", "meaning": "second number of the second pair" } ], "sympy": "Eq(a*d, b*c)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/proportion" ] }, { "id": "ball-mathematical-recreations-1905/eq-62cd19e4d0", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 41", "location": "Some Arithmetical Questions", "latex": "\\phi=(x^2-y^2)/(x^2 + y^2)^2", "name": null, "statement": "Defines the function phi of x and y used in the integral-order fallacy (footnote to the Fallacies section).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function of x and y integrated in the example" }, { "unit": null, "symbol": "x", "meaning": "integration variable" }, { "unit": null, "symbol": "y", "meaning": "integration variable" } ], "sympy": "Eq(phi, (x**2 - y**2)/(x**2 + y**2)**2)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/integral" ] }, { "id": "ball-mathematical-recreations-1905/eq-917697a68e", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 41", "location": "Some Arithmetical Questions", "latex": "\\frac{1}{4}\\pi =-\\frac{1}{4}\\pi", "name": null, "statement": "The book states that the iterated integrals of phi in the two orders give pi/4 and -pi/4 respectively, which is the claimed contradiction (footnote; Bertrand).", "kind": "result", "symbols": [ { "unit": null, "symbol": "pi", "meaning": "the circle constant" } ], "sympy": "Eq(pi/4, -pi/4)", "physics": false, "states": [], "concepts": [ "concept/arithmetical-fallacy", "concept/integral", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-6438c8668e", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 33", "location": "Some Arithmetical Questions", "latex": "y < n-m", "name": null, "statement": "The number of cards transferred from bottom to top must be less than n - m.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "y", "meaning": "number of cards transferred from the bottom to the top of the pack" }, { "unit": null, "symbol": "n", "meaning": "number of cards in the pack" }, { "unit": null, "symbol": "m", "meaning": "number of top cards reversed" } ], "sympy": "Lt(y, n - m)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/rule" ] }, { "id": "ball-mathematical-recreations-1905/eq-d9011b5830", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 32", "location": "Some Arithmetical Questions", "latex": "m<12", "name": null, "statement": "The starting hour m is less than 12, so n + 12 - m is always positive.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "m", "meaning": "the hour the spectator thinks of" } ], "sympy": "Lt(m, 12)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/rule" ] }, { "id": "ball-mathematical-recreations-1905/eq-059b1071b2", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 33", "location": "Some Arithmetical Questions", "latex": "m < 20", "name": null, "statement": "The card or domino trick works for any collection of m distinguishable things provided m < 20.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "m", "meaning": "number of distinguishable things arranged on the table" } ], "sympy": "Lt(m, 20)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/rule" ] }, { "id": "ball-mathematical-recreations-1905/eq-24e48116c8", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 46", "location": "Some Arithmetical Questions", "latex": "(m + n) ! / m! n !", "name": null, "statement": "The number of routes from the top left-hand corner to the bottom right-hand corner of a board of m by n cells, moving only downward or rightward along the ruled lines.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "m", "meaning": "number of cells across the board (the board has m+1 vertical ruled lines)" }, { "unit": null, "symbol": "n", "meaning": "number of cells down the board (the board has n+1 horizontal ruled lines)" } ], "sympy": "Eq(R, factorial(m + n)/(factorial(m)*factorial(n)))", "physics": false, "states": [], "concepts": [ "concept/factor", "concept/lattice-path-counting", "concept/route" ] }, { "id": "ball-mathematical-recreations-1905/eq-6eb1898e08", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 47", "location": "Some Arithmetical Questions", "latex": "(52!)/(13!)^4", "name": null, "statement": "The number of possible distributions of hands at whist from a pack of fifty-two cards, given as a footnote.", "kind": "formula", "symbols": [], "sympy": "Eq(N, factorial(52)/(factorial(13)**4))", "physics": false, "states": [], "concepts": [ "concept/permutation" ] }, { "id": "ball-mathematical-recreations-1905/eq-eeb3450d00", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 47", "location": "Some Arithmetical Questions", "latex": "pr/(ns + r)", "name": null, "statement": "Under the cumulative vote, the least number of supporters who can secure a candidate's election must exceed this quantity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "number of electors" }, { "unit": null, "symbol": "r", "meaning": "number of votes each elector has" }, { "unit": null, "symbol": "s", "meaning": "maximum number of an elector's votes that may be given to one candidate" }, { "unit": null, "symbol": "n", "meaning": "number of men to be elected" }, { "unit": null, "symbol": "S", "meaning": "least number of supporters who can secure the election of a candidate" } ], "sympy": "Gt(S, p*r/(n*s + r))", "physics": false, "states": [], "concepts": [ "concept/cumulative-voting", "concept/inequality", "concept/minority-representation" ] }, { "id": "ball-mathematical-recreations-1905/eq-c5e702b7eb", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 47", "location": "Some Arithmetical Questions", "latex": "na/(n + 1)", "name": null, "statement": "The man who reaches the greatest distance into the desert occupies this many days before he returns to the starting point.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of explorers" }, { "unit": "day", "symbol": "a", "meaning": "number of days that one man's provisions last" }, { "unit": "day", "symbol": "T", "meaning": "days occupied by the furthest explorer before returning to the start" } ], "sympy": "Eq(T, n*a/(n + 1))", "physics": false, "states": [], "concepts": [ "concept/exploration-problem" ] }, { "id": "ball-mathematical-recreations-1905/eq-c7a3606eff", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 47", "location": "Some Arithmetical Questions", "latex": "\\frac{1}{2}a (1 + \\frac{1}{2} + \\frac{1}{3}+ \\dotsb + 1/n)", "name": null, "statement": "If the explorers may make depots, the longest possible journey occupies this many days.", "kind": "result", "symbols": [ { "unit": "day", "symbol": "a", "meaning": "number of days that one man's provisions last" }, { "unit": null, "symbol": "n", "meaning": "number of explorers" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/exploration-problem", "concept/infinite-sequence", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-4ad76a73b2", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 49", "location": "Some Arithmetical Questions", "latex": "\\frac{1}{2}(3^n-1)", "name": null, "statement": "With weights of 1, 3, 3^2, ..., 3^(n-1) pounds one can weigh every integral number of pounds from 1 up to this many pounds, which is the least number of weights for the problem.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of weights in the series 1, 3, 3^2, ..., 3^(n-1)" }, { "unit": "pound", "symbol": "W", "meaning": "greatest integral number of pounds that can be weighed" } ], "sympy": "Eq(W, (3**n - 1)/2)", "physics": false, "states": [], "concepts": [ "concept/bachet-s-weights-problem", "concept/exponent", "concept/power", "concept/ternary-scale-of-notation" ] }, { "id": "ball-mathematical-recreations-1905/eq-52778a0a03", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 50", "location": "Some Arithmetical Questions", "latex": "(1-x^{81})/x^{40} (1-x)", "name": null, "statement": "The sum x^(-40) + x^(-39) + ... + 1 + ... + x^40 is equal to this expression, which can be factored in the way MacMahon used for Bachet's problem.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable of the polynomial in Laurent form" }, { "unit": null, "symbol": "S", "meaning": "the sum x^(-40) + x^(-39) + ... + x^40" } ], "sympy": "Eq(S, (1 - x**81)/(x**40*(1 - x)))", "physics": false, "states": [], "concepts": [ "concept/bachet-s-weights-problem", "concept/exponent", "concept/factor", "concept/power" ] }, { "id": "ball-mathematical-recreations-1905/eq-eea7c7cb02", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 51", "location": "Some Arithmetical Questions", "latex": "2^p-1", "name": "Mersenne number", "statement": "The number 2^p - 1, which Mersenne's rule tests for primality for various exponents p.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "p", "meaning": "exponent (a prime index in Mersenne's rule)" } ], "sympy": "Eq(M, 2**p - 1)", "physics": false, "states": [ "concept/mersenne-number" ], "concepts": [ "concept/exponent", "concept/prime-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-b72429d07a", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 52", "location": "Some Arithmetical Questions", "latex": "2^{p-1} (2^p-1)", "name": null, "statement": "All perfect numbers are believed to be included in this formula, where 2^p - 1 is prime; Euclid proved that every number of this form is perfect and Euler showed it includes all even perfect numbers.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "exponent such that 2^p - 1 is prime" }, { "unit": null, "symbol": "P", "meaning": "perfect number" } ], "sympy": "Eq(P, 2**(p-1)*(2**p - 1))", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/mersenne-number", "concept/perfect-number", "concept/prime-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-a50dcc891e", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 53", "location": "Some Arithmetical Questions", "latex": "2^m + 1", "name": "Fermat number", "statement": "Fermat asserted that numbers of this form are prime when m = 2^n; the assertion is false, as Euler showed for n = 5.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "m", "meaning": "exponent, here a power of 2 (m = 2^n) in Fermat's theorem on binary powers" }, { "unit": null, "symbol": "F", "meaning": "Fermat number" } ], "sympy": "Eq(F, 2**m + 1)", "physics": false, "states": [ "concept/fermat-number" ], "concepts": [ "concept/exponent", "concept/prime-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-ada03c56fe", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 54", "location": "Some Arithmetical Questions", "latex": "x^n + y^n = z^n", "name": "Fermat's Last Theorem", "statement": "No integral values of x, y, z satisfy this equation when n is an integer greater than 2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "integer unknown" }, { "unit": null, "symbol": "y", "meaning": "integer unknown" }, { "unit": null, "symbol": "z", "meaning": "integer unknown" }, { "unit": null, "symbol": "n", "meaning": "integer exponent greater than 2" } ], "sympy": "Eq(x**n + y**n, z**n)", "physics": false, "states": [ "theorem/fermat-s-last-theorem" ], "concepts": [ "concept/exponent", "concept/integer", "concept/power", "concept/unknown" ] }, { "id": "ball-mathematical-recreations-1905/eq-8843e7b50a", "chapter": "ball-mathematical-recreations-1905/ch-i", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 57", "location": "Some Arithmetical Questions", "latex": "x^2 + 2 = y^3", "name": null, "statement": "Fermat's problem to show that this equation has only one integral solution, namely x = 5, y = 3.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown integer" }, { "unit": null, "symbol": "y", "meaning": "unknown integer" } ], "sympy": "Eq(x**2 + 2, y**3)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/power", "concept/solution", "concept/unknown" ] }, { "id": "ball-mathematical-recreations-1905/eq-a5f1024497", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 67", "location": "Some Geometrical Questions", "latex": "5 \\times 13 - 8^2 = 1", "name": null, "statement": "The integer relation on which the paradox of the 64-square board that yields 65 squares depends; similar identities follow from the continued-fraction convergents.", "kind": "identity", "symbols": [], "sympy": "Eq(5*13 - 8**2, 1)", "physics": false, "states": [], "concepts": [ "concept/dissection-proof", "concept/geometrical-paradox", "concept/integer" ] }, { "id": "ball-mathematical-recreations-1905/eq-3de0f1b3c4", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 65", "location": "Some Geometrical Questions", "latex": "\\frac{1}{2}\\pi ab", "name": null, "statement": "The area of the semi-ellipse bounded by the minor axis equals one half of pi times a times b, whatever the dimensions of the curve.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "area of the semi-ellipse bounded by the minor axis" }, { "unit": null, "symbol": "a", "meaning": "semi-axis of the ellipse (usual notation; not defined in this chapter)" }, { "unit": null, "symbol": "b", "meaning": "semi-axis of the ellipse (usual notation; not defined in this chapter)" } ], "sympy": "Eq(A, pi*a*b/2)", "physics": false, "states": [], "concepts": [ "concept/area", "concept/ellipse", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-aea449f253", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 72", "location": "Some Geometrical Questions", "latex": "h &=1 + p_1 + 2p_2 + \\dotsb\\,", "name": null, "statement": "The number of hills equals one plus the number of single passes plus twice the number of double passes, and so on, by the theorem of Cauchy and Euler.", "kind": "result", "symbols": [ { "unit": null, "symbol": "h", "meaning": "number of hills (equal to the number of summits)" }, { "unit": null, "symbol": "p_1", "meaning": "number of single passes" }, { "unit": null, "symbol": "p_2", "meaning": "number of double passes" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/contour-line", "concept/hill", "concept/pass", "concept/summit", "person/augustin-louis-cauchy", "person/leonhard-euler" ] }, { "id": "ball-mathematical-recreations-1905/eq-9e217b298e", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 72", "location": "Some Geometrical Questions", "latex": "d &=1 + f_1 + 2f_2 + \\dotsb\\,", "name": null, "statement": "The number of dales equals one plus the number of single forks plus twice the number of double forks, and so on.", "kind": "result", "symbols": [ { "unit": null, "symbol": "d", "meaning": "number of dales (equal to the number of bottoms)" }, { "unit": null, "symbol": "f_1", "meaning": "number of single forks" }, { "unit": null, "symbol": "f_2", "meaning": "number of double forks" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/contour-line", "concept/dale", "concept/fork", "person/augustin-louis-cauchy", "person/leonhard-euler" ] }, { "id": "ball-mathematical-recreations-1905/eq-a7a816d5d8", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 72", "location": "Some Geometrical Questions", "latex": "w &=2(p_1 + f_1) + 3(p_2 + f_2)) + \\dotsb\\,", "name": null, "statement": "The number of watercourses equals twice the sum of single passes and single forks, plus three times the sum of double passes and double forks, and so on.", "kind": "result", "symbols": [ { "unit": null, "symbol": "w", "meaning": "number of watercourses (equal to the number of watersheds)" }, { "unit": null, "symbol": "p_1", "meaning": "number of single passes" }, { "unit": null, "symbol": "f_1", "meaning": "number of single forks" }, { "unit": null, "symbol": "p_2", "meaning": "number of double passes" }, { "unit": null, "symbol": "f_2", "meaning": "number of double forks" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/contour-line", "concept/fork", "concept/pass", "concept/watercourse", "concept/watershed", "person/augustin-louis-cauchy", "person/leonhard-euler" ] }, { "id": "ball-mathematical-recreations-1905/eq-afeb08ebcd", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 81", "location": "Some Geometrical Questions", "latex": "y = 2n - 1", "name": null, "statement": "For n married couples with a boat carrying x = 4 people (n > 5), the least number of passages y from one bank to the other is 2n - 1 (Delannoy's result).", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "least number of passages from one bank to the other" }, { "unit": null, "symbol": "n", "meaning": "number of married couples" } ], "sympy": "Eq(y, 2*n - 1)", "physics": false, "states": [], "concepts": [ "concept/ferry-boat-problems", "concept/integer-part", "concept/minimum" ] }, { "id": "ball-mathematical-recreations-1905/eq-d5a055f8da", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 75", "location": "Some Geometrical Questions", "latex": "x=\\pm a", "name": null, "statement": "One of the ten lines whose intersection points, placed as counters, give the 19-counter arrangement in 10 rows of five; the lines are x = ±a, x = ±b, y = ±a, y = ±b, y = ±x.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "horizontal coordinate of a point in the plane" }, { "unit": null, "symbol": "a", "meaning": "a constant (a distance from the origin along an axis)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/counters-in-a-row", "concept/line" ] }, { "id": "ball-mathematical-recreations-1905/eq-5c6bb2edf4", "chapter": "ball-mathematical-recreations-1905/ch-ii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 75", "location": "Some Geometrical Questions", "latex": "y=\\pm x", "name": null, "statement": "The diagonal pair of lines through the origin; one of the ten lines used in the 19-counter, 10-row construction.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "horizontal coordinate of a point in the plane" }, { "unit": null, "symbol": "y", "meaning": "vertical coordinate of a point in the plane" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/line" ] }, { "id": "ball-mathematical-recreations-1905/eq-bc9efdd699", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 98", "location": "Some Mechanical Questions", "latex": "a + a/n + a/n^2 + a/n^3 + \\dotsb", "name": null, "statement": "The time for a point moving uniformly along the equiangular spiral to reach the pole is the infinite sum of the times for successive convolutions, each 1/n of the one before.", "kind": "result", "symbols": [ { "unit": "seconds", "symbol": "a", "meaning": "time taken to cover the first convolution of the spiral" }, { "unit": null, "symbol": "n", "meaning": "constant ratio by which each convolution is shorter than the one outside it (each is 1/nth of that one)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/equiangular-spiral", "concept/geometrical-progression", "concept/infinite-set", "concept/sum", "concept/uniform-motion" ] }, { "id": "ball-mathematical-recreations-1905/eq-51a1332c94", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 98", "location": "Some Mechanical Questions", "latex": "an/(n-1)", "name": null, "statement": "The closed-form value of the infinite geometric sum, so the point reaches the pole in a finite time an/(n-1) seconds despite circling it infinitely many times.", "kind": "result", "symbols": [ { "unit": "seconds", "symbol": "a", "meaning": "time taken to cover the first convolution of the spiral" }, { "unit": null, "symbol": "n", "meaning": "constant ratio by which each convolution is shorter than the one outside it (each is 1/nth of that one)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/geometrical-progression", "concept/infinity", "concept/sum", "concept/uniform-motion", "method/limiting-case" ] }, { "id": "ball-mathematical-recreations-1905/eq-d725ea4f54", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "W\\!AR = \\theta", "name": null, "statement": "The angle between the wind direction WA and the keel AR is called theta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "angle between the wind direction and the keel" } ], "sympy": "Eq(WAR, theta)", "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/sailing-faster-than-the-wind" ] }, { "id": "ball-mathematical-recreations-1905/eq-af74ae2ee4", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "BAS = \\alpha", "name": null, "statement": "The angle between the keel AB and the sail AS is called alpha.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" } ], "sympy": "Eq(BAS, alpha)", "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/sailing-faster-than-the-wind" ] }, { "id": "ball-mathematical-recreations-1905/eq-67a2299902", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "W\\!AL=\\theta + \\alpha", "name": null, "statement": "The angle between the wind direction and the sail AL equals theta plus alpha.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "angle between the wind direction and the keel" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" } ], "sympy": "Eq(WAL, theta + alpha)", "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/sailing-faster-than-the-wind" ] }, { "id": "ball-mathematical-recreations-1905/eq-f5c8e2442a", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "v \\sin \\alpha = u \\sin (\\theta + \\alpha)", "name": null, "statement": "For steady motion of the boat, the boat speed times the sine of the sail angle equals the wind speed times the sine of the sum of the wind and sail angles, so that the resultant pressure normal to the sail vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "velocity of the boat in the direction AB" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" }, { "unit": null, "symbol": "u", "meaning": "velocity of the wind" }, { "unit": null, "symbol": "θ", "meaning": "angle the wind direction makes with the keel" } ], "sympy": "Eq(v*sin(alpha), u*sin(theta + alpha))", "physics": true, "states": [], "concepts": [ "concept/plane-angle", "concept/relative-motion", "concept/sailing-faster-than-the-wind", "concept/sine", "concept/steady-motion", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/eq-0d3c3a6367", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "v>u", "name": null, "statement": "The boat moves faster than the wind when sin(theta + alpha) is greater than sin alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "velocity of the boat" }, { "unit": null, "symbol": "u", "meaning": "velocity of the wind" } ], "sympy": "Gt(v, u)", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/sailing-faster-than-the-wind", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/eq-fb7ad75362", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "\\sin (\\theta + \\alpha)>\\allowbreak\\sin \\alpha", "name": null, "statement": "If sin(theta + alpha) is greater than sin alpha, then the boat's speed exceeds the wind speed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "angle the wind direction makes with the keel" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" } ], "sympy": "Gt(sin(theta + alpha), sin(alpha))", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/sailing-faster-than-the-wind", "concept/sine" ] }, { "id": "ball-mathematical-recreations-1905/eq-280fc5f950", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "\\theta + \\alpha = \\frac{1}{2}\\pi", "name": null, "statement": "With the sail angle fixed, the boat speed is a maximum when theta plus alpha is a right angle, i.e. theta is the complement of alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "angle the wind direction makes with the keel" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel (constant)" } ], "sympy": "Eq(theta + alpha, pi/2)", "physics": true, "states": [], "concepts": [ "concept/complementary-angles", "concept/maximum", "concept/sailing-faster-than-the-wind", "quantity/right-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-0ef6b17d24", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "v = u \\cosec \\alpha", "name": null, "statement": "At the maximum, the boat speed equals the wind speed times the cosecant of the sail angle, which is greater than the wind speed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "velocity of the boat" }, { "unit": null, "symbol": "u", "meaning": "velocity of the wind" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" } ], "sympy": "Eq(v, u/sin(alpha))", "physics": true, "states": [], "concepts": [ "concept/cosecant", "concept/maximum", "concept/sailing-faster-than-the-wind", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/eq-dc2ec1c628", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "v \\sin \\alpha = u \\sin \\phi", "name": null, "statement": "For a boat running close to the wind, the steady-motion condition is written with phi, the angle WAS between the wind and the sail.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "velocity of the boat" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" }, { "unit": null, "symbol": "u", "meaning": "velocity of the wind" }, { "unit": null, "symbol": "φ", "meaning": "angle WAS between the wind direction and the sail" } ], "sympy": "Eq(v*sin(alpha), u*sin(phi))", "physics": true, "states": [], "concepts": [ "concept/plane-angle", "concept/sailing-faster-than-the-wind", "concept/sine", "concept/steady-motion" ] }, { "id": "ball-mathematical-recreations-1905/eq-653c731ed4", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "\\phi = \\text{angle } W\\!AS = \\pi - \\theta - \\alpha", "name": null, "statement": "The angle phi between the wind and the sail is pi minus theta minus alpha.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "angle WAS between the wind direction and the sail" }, { "unit": null, "symbol": "θ", "meaning": "angle the wind direction makes with the keel" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" } ], "sympy": "Eq(phi, pi - theta - alpha)", "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/sailing-faster-than-the-wind" ] }, { "id": "ball-mathematical-recreations-1905/eq-253a2ddc86", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "v = u \\sin \\phi \\cosec \\alpha", "name": null, "statement": "For a boat running close to the wind, the boat speed is the wind speed times sin phi times the cosecant of alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "velocity of the boat" }, { "unit": null, "symbol": "u", "meaning": "velocity of the wind" }, { "unit": null, "symbol": "φ", "meaning": "angle WAS between the wind direction and the sail" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" } ], "sympy": "Eq(v, u*sin(phi)/sin(alpha))", "physics": true, "states": [], "concepts": [ "concept/cosecant", "concept/sailing-faster-than-the-wind", "concept/sine", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/eq-6d58cd09ed", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "w =\\allowbreak v \\cos BAW =\\allowbreak v \\cos (\\alpha + \\phi) =\\allowbreak u \\sin \\phi \\cosec \\alpha \\cos (\\alpha + \\phi)", "name": null, "statement": "The component velocity of the boat in the teeth of the wind equals the boat speed times the cosine of the angle BAW, which is alpha plus phi, giving w in terms of u, phi and alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "w", "meaning": "component velocity of the boat in the teeth of the wind (in the direction AW)" }, { "unit": null, "symbol": "v", "meaning": "velocity of the boat" }, { "unit": null, "symbol": "u", "meaning": "velocity of the wind" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" }, { "unit": null, "symbol": "φ", "meaning": "angle WAS between the wind direction and the sail" } ], "sympy": "Eq(w, v*cos(alpha + phi))", "physics": true, "states": [], "concepts": [ "concept/cosecant", "concept/cosine", "concept/resultant", "concept/sailing-faster-than-the-wind", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/eq-262775e3ed", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "\\phi = \\frac{1}{4}\\pi - \\frac{1}{2}\\alpha", "name": null, "statement": "With alpha constant, the component velocity w in the teeth of the wind is a maximum when phi equals a quarter of pi minus half of alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "angle WAS between the wind direction and the sail" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel (constant)" } ], "sympy": "Eq(phi, pi/4 - alpha/2)", "physics": true, "states": [], "concepts": [ "concept/maximum", "concept/plane-angle", "concept/sailing-faster-than-the-wind" ] }, { "id": "ball-mathematical-recreations-1905/eq-4b65af3edc", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "w = \\frac{1}{2}u (\\cosec\\alpha - 1)", "name": null, "statement": "At the maximum, the component velocity of the boat in the teeth of the wind is half the wind speed times (cosecant alpha minus 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "w", "meaning": "component velocity of the boat in the teeth of the wind" }, { "unit": null, "symbol": "u", "meaning": "velocity of the wind" }, { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" } ], "sympy": "Eq(w, u*(1/sin(alpha) - 1)/2)", "physics": true, "states": [], "concepts": [ "concept/cosecant", "concept/maximum", "concept/sailing-faster-than-the-wind", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/eq-1e7ba35e99", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 111", "location": "Some Mechanical Questions", "latex": "\\sin\\alpha< \\frac{1}{3}", "name": null, "statement": "The component velocity in the teeth of the wind is greater than the wind speed when sin alpha is less than one third.", "kind": "result", "symbols": [ { "unit": null, "symbol": "α", "meaning": "angle the sail makes with the keel" } ], "sympy": "Lt(sin(alpha), Rational(1, 3))", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/sailing-faster-than-the-wind", "concept/sine" ] }, { "id": "ball-mathematical-recreations-1905/eq-aefcc06c6e", "chapter": "ball-mathematical-recreations-1905/ch-iii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 113", "location": "Some Mechanical Questions", "latex": "p = \\Pi\\alpha^{-v^2}", "name": "Hauksbee's law", "statement": "In an elastic perfect fluid whose pressure is proportional to its density, the pressure falls exponentially with the square of the steady velocity of the fluid.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "Π", "meaning": "constant" }, { "unit": null, "symbol": "α", "meaning": "constant" }, { "unit": null, "symbol": "v", "meaning": "steady velocity of the fluid" } ], "sympy": "Eq(p, Pi*alpha**(-v**2))", "physics": true, "states": [ "law/hauksbee-s-law" ], "concepts": [ "concept/constant", "concept/motion-of-fluids", "concept/pressure", "concept/steady-motion", "quantity/velocity" ] }, { "id": "ball-mathematical-recreations-1905/eq-5bf540796b", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 125", "location": "Some Miscellaneous Questions", "latex": "2^{64}-1", "name": null, "statement": "The number of single-disc transfers needed to move the Tower of Hanoi with sixty-four discs is 2 to the 64th power minus 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "T", "meaning": "number of separate transfers of single discs" }, { "unit": null, "symbol": "n", "meaning": "number of discs in the tower" } ], "sympy": "Eq(T, 2**64 - 1)", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/operation", "concept/tower-of-hanoi" ] }, { "id": "ball-mathematical-recreations-1905/eq-78c46b5717", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 125", "location": "Some Miscellaneous Questions", "latex": "2^n-1", "name": null, "statement": "A tower of n discs needs 2 to the power n minus 1 single transfers, so eight discs need 255.", "kind": "result", "symbols": [ { "unit": null, "symbol": "T", "meaning": "number of separate transfers of single discs" }, { "unit": null, "symbol": "n", "meaning": "number of discs in the tower" } ], "sympy": "Eq(T, 2**n - 1)", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/operation", "concept/tower-of-hanoi" ] }, { "id": "ball-mathematical-recreations-1905/eq-c0b66baf82", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 124", "location": "Some Miscellaneous Questions", "latex": "2x +1", "name": null, "statement": "Moving a tower of n discs takes twice the x transfers needed for n-1 discs, plus one, by the recursive method.", "kind": "result", "symbols": [ { "unit": null, "symbol": "T", "meaning": "number of separate transfers of single discs for a tower of n discs" }, { "unit": null, "symbol": "x", "meaning": "number of transfers needed for a tower of n-1 discs" } ], "sympy": "Eq(T, 2*x + 1)", "physics": false, "states": [], "concepts": [ "concept/formula", "concept/operation", "concept/tower-of-hanoi" ] }, { "id": "ball-mathematical-recreations-1905/eq-fbc382aaa8", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 127", "location": "Some Miscellaneous Questions", "latex": "\\frac{1}{3}(2^{n+1}-1)", "name": null, "statement": "For an odd number n of Chinese rings, disconnecting them from the bar takes one third of (2 to the power n+1 minus 1) steps.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of steps (taking a ring off or putting one on the bar)" }, { "unit": null, "symbol": "n", "meaning": "number of rings" } ], "sympy": "Eq(S, Rational(1,3)*(2**(n+1) - 1))", "physics": false, "states": [], "concepts": [ "concept/binary-scale", "concept/chinese-rings", "concept/exponent", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-de4fad9a9b", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 127", "location": "Some Miscellaneous Questions", "latex": "\\frac{1}{3}(2^{n+1}-2)", "name": null, "statement": "For an even number n of Chinese rings, disconnecting them from the bar takes one third of (2 to the power n+1 minus 2) steps.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of steps (taking a ring off or putting one on the bar)" }, { "unit": null, "symbol": "n", "meaning": "number of rings" } ], "sympy": "Eq(S, Rational(1,3)*(2**(n+1) - 2))", "physics": false, "states": [], "concepts": [ "concept/chinese-rings", "concept/exponent", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-3df447e0eb", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 128", "location": "Some Miscellaneous Questions", "latex": "\\tfrac{1}{3}(2^{2n+2} - 1)", "name": null, "statement": "The number of steps to take off the first 2n+1 Chinese rings is one third of (2 to the power 2n+2 minus 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of steps needed to take off the first rings" }, { "unit": null, "symbol": "n", "meaning": "half of (number of rings minus 1), so the rings number 2n+1" } ], "sympy": "Eq(S, (2**(2*n+2) - 1)/3)", "physics": false, "states": [], "concepts": [ "concept/chinese-rings", "concept/exponent", "concept/sum", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-9ad40513e9", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 128", "location": "Some Miscellaneous Questions", "latex": "\\tfrac{1}{3}(2^{2n+1}-2)", "name": null, "statement": "The number of steps to take off the first 2n Chinese rings is one third of (2 to the power 2n+1 minus 2).", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of steps needed to take off the first rings" }, { "unit": null, "symbol": "n", "meaning": "half the number of rings taken off" } ], "sympy": "Eq(S, (2**(2*n+1) - 2)/3)", "physics": false, "states": [], "concepts": [ "concept/chinese-rings", "concept/exponent", "concept/sum", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-060de4ea59", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 128", "location": "Some Miscellaneous Questions", "latex": "2^{2n}", "name": null, "statement": "If the first two rings are taken off or put on in one step, the count for an even number of rings becomes 2 to the power 2n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of steps when the first two rings move together" }, { "unit": null, "symbol": "n", "meaning": "half the number of rings" } ], "sympy": "Eq(S, 2**(2*n))", "physics": false, "states": [], "concepts": [ "concept/chinese-rings", "concept/exponent", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-0f2d319ac4", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 128", "location": "Some Miscellaneous Questions", "latex": "2^{2n-1}-1", "name": null, "statement": "If the first two rings are taken off or put on in one step, the count for an odd number of rings becomes 2 to the power 2n-1 minus 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of steps when the first two rings move together" }, { "unit": null, "symbol": "n", "meaning": "parameter of the ring count" } ], "sympy": "Eq(S, 2**(2*n - 1) - 1)", "physics": false, "states": [], "concepts": [ "concept/chinese-rings", "concept/exponent", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-ac74f8ab5b", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 128", "location": "Some Miscellaneous Questions", "latex": "4x", "name": null, "statement": "Once the first m rings are off, the additional steps needed to take off the (m+3)th and (m+4)th rings equal 4x, where x is the step count for the previous pair.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "additional steps needed to take off the next two rings" }, { "unit": null, "symbol": "x", "meaning": "steps needed to take off the (m+1)th and (m+2)th rings from the position with the first m rings off" } ], "sympy": "Eq(S, 4*x)", "physics": false, "states": [], "concepts": [ "concept/chinese-rings", "concept/recursion", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-b6431073bf", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 128", "location": "Some Miscellaneous Questions", "latex": "1 + 4 + 4^2 + \\dotsb + 4^n", "name": null, "statement": "The steps needed to take off the first 2n+1 rings are the sum of the powers of 4 from 4 to the power 0 up to 4 to the power n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "total steps to take off the first 2n+1 rings" }, { "unit": null, "symbol": "n", "meaning": "index of the largest power of 4 in the sum" } ], "sympy": "Eq(S, Sum(4**k, (k, 0, n)))", "physics": false, "states": [], "concepts": [ "concept/chinese-rings", "concept/exponent", "concept/sum", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-47454ee531", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 130", "location": "Some Miscellaneous Questions", "latex": "(1+2^1 + 2^2 + \\ldots + 2^{2n})", "name": null, "statement": "Putting on a set of 2n+1 rings in the binary numbering requires the sum of the powers of 2 from 2 to the power 0 up to 2 to the power 2n steps.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of steps to put on a set of rings" }, { "unit": null, "symbol": "n", "meaning": "half of (number of rings minus 1)" } ], "sympy": "Eq(S, Sum(2**k, (k, 0, 2*n)))", "physics": false, "states": [], "concepts": [ "concept/binary-scale", "concept/chinese-rings", "concept/exponent", "concept/sum", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-46183fb45e", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 130", "location": "Some Miscellaneous Questions", "latex": "(2 + 2^3 + \\ldots + 2^{2n-1})", "name": null, "statement": "Putting on a set of 2n rings in the binary numbering requires the sum of the even powers of 2 starting at 2 up to 2 to the power 2n-1 steps.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of steps to put on a set of rings" }, { "unit": null, "symbol": "n", "meaning": "half the number of rings" } ], "sympy": "Eq(S, Sum(2**(2*k+1), (k, 0, n-1)))", "physics": false, "states": [], "concepts": [ "concept/binary-scale", "concept/chinese-rings", "concept/exponent", "concept/sum", "concept/vector" ] }, { "id": "ball-mathematical-recreations-1905/eq-f8df9b3cef", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 123", "location": "Some Miscellaneous Questions", "latex": "mn-3", "name": null, "statement": "For a rectangular box with both m and n even, a right-angle rotation is equivalent to mn-3 simple interchanges, so it changes a solvable position into an unsolvable one and vice versa.", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "number of simple interchanges equivalent to a rotation of the box" }, { "unit": null, "symbol": "m", "meaning": "number of rows of cells in the rectangular box" }, { "unit": null, "symbol": "n", "meaning": "number of columns of cells in the rectangular box" } ], "sympy": "Eq(N, m*n - 3)", "physics": false, "states": [], "concepts": [ "concept/fifteen-puzzle", "concept/interchange", "concept/permutation", "concept/rotation" ] }, { "id": "ball-mathematical-recreations-1905/eq-51aeea8ffa", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 121", "location": "Some Miscellaneous Questions", "latex": "n-1", "name": null, "statement": "A cyclical permutation of n letters is equivalent to n-1 simple interchanges.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "N", "meaning": "number of simple interchanges equivalent to the cyclical permutation" }, { "unit": null, "symbol": "n", "meaning": "number of letters (counters) in the cyclical permutation" } ], "sympy": "Eq(N, n - 1)", "physics": false, "states": [], "concepts": [ "concept/cyclical-permutation", "concept/interchange", "concept/permutation" ] }, { "id": "ball-mathematical-recreations-1905/eq-3f97b4225d", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "y=(p^m-1)/(p-1)", "name": null, "statement": "If p is a prime and x = p^m, a = p, b = 2, then the greatest number of ways y is (p^m - 1)/(p - 1) (Kirkman's theorem).", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "a prime number" }, { "unit": null, "symbol": "m", "meaning": "exponent with x = p^m" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, (p**m-1)/(p-1))", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/kirkman-s-schoolgirl-problem", "concept/prime-number", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-edcd10b5b0", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 137", "location": "Some Miscellaneous Questions", "latex": "\\frac{1}{2}(n-1)(n-2) (n^2 + 3n-2)", "name": null, "statement": "The number of ways to place two kings on an n-by-n board (n^2 cells) so that they do not occupy adjacent squares is said to be this expression (an assertion the book reports, not proves).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "side of the board, which has n^2 cells" } ], "sympy": "Eq(N, (n-1)*(n-2)*(n**2+3*n-2)/2)", "physics": false, "states": [], "concepts": [ "concept/chess-piece-placement-problem", "concept/permutation", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-28b1069961", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 137", "location": "Some Miscellaneous Questions", "latex": "\\frac{1}{6}(n-1) (n-2) (n^4+ 3n^3-20n^2-30n + 132)", "name": null, "statement": "The number of ways to place three kings on an n-by-n board so that no two occupy adjacent squares is said to be this expression (reported as an assertion, not proved).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "side of the board, which has n^2 cells" } ], "sympy": "Eq(N, (n-1)*(n-2)*(n**4+3*n**3-20*n**2-30*n+132)/6)", "physics": false, "states": [], "concepts": [ "concept/chess-piece-placement-problem", "concept/permutation", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-ccca12397a", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 146", "location": "Some Miscellaneous Questions", "latex": "x_1=\\frac{1}{2}(2p+x_0+1)", "name": null, "statement": "One shuffle of a pack of 2p cards moves the card in place x_0 (x_0 odd) to place x_1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "half the number of cards in the pack (pack has 2p cards)" }, { "unit": null, "symbol": "x_0", "meaning": "initial position of the card (odd)" }, { "unit": null, "symbol": "x_1", "meaning": "position of the card after one shuffle" } ], "sympy": "Eq(x_1, (2*p+x_0+1)/2)", "physics": false, "states": [], "concepts": [ "concept/arrangement", "concept/permutation", "method/card-shuffling" ] }, { "id": "ball-mathematical-recreations-1905/eq-3299123e29", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 146", "location": "Some Miscellaneous Questions", "latex": "x_1=\\frac{1}{2}(2p-x_0 + 2)", "name": null, "statement": "One shuffle of a pack of 2p cards moves the card in place x_0 (x_0 even) to place x_1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "half the number of cards in the pack (pack has 2p cards)" }, { "unit": null, "symbol": "x_0", "meaning": "initial position of the card (even)" }, { "unit": null, "symbol": "x_1", "meaning": "position of the card after one shuffle" } ], "sympy": "Eq(x_1, (2*p-x_0+2)/2)", "physics": false, "states": [], "concepts": [ "concept/arrangement", "concept/permutation", "method/card-shuffling" ] }, { "id": "ball-mathematical-recreations-1905/eq-d68fe73cf6", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 146", "location": "Some Miscellaneous Questions", "latex": "2^{m+1}x_m=(4p+1)(2^{m-1} \\pm 2^{m-2} \\pm \\dotsb \\pm 2 \\pm 1) \\pm 2x_0 + 2^m \\pm 1", "name": null, "statement": "After m shuffles of a pack of 2p cards, the card starting in place x_0 ends in place x_m, where the signs are an ambiguity of sign.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "half the number of cards in the pack (pack has 2p cards)" }, { "unit": null, "symbol": "m", "meaning": "number of shuffles" }, { "unit": null, "symbol": "x_0", "meaning": "initial position of the card" }, { "unit": null, "symbol": "x_m", "meaning": "position of the card after m shuffles" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/arrangement", "concept/cyclical-permutation", "concept/permutation", "method/card-shuffling" ] }, { "id": "ball-mathematical-recreations-1905/eq-233d9f8903", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 141", "location": "Some Miscellaneous Questions", "latex": "3 (2m+1)(3m+1)", "name": null, "statement": "The number of fundamental Anstician arrangements is this expression in m.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "integer with p = 12m+7 prime, so there are 2m+1 girls-in-triplets parameter for 2p+1 girls" } ], "sympy": "Eq(N, 3*(2*m+1)*(3*m+1))", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number", "concept/triplet", "method/anstice-s-method" ] }, { "id": "ball-mathematical-recreations-1905/eq-0cb131f730", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "y=n(2n-1)", "name": null, "statement": "For x = 2n girls walking in rows of 2 with every pair together once, the greatest number of ways y is n(2n-1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls" }, { "unit": null, "symbol": "a", "meaning": "number of girls walking abreast in each row" }, { "unit": null, "symbol": "b", "meaning": "size of the combination of girls required to walk together once" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" }, { "unit": null, "symbol": "z", "meaning": "number of days in the solution" } ], "sympy": "Eq(y, n*(2*n-1))", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number", "concept/solution" ] }, { "id": "ball-mathematical-recreations-1905/eq-1340dc0e2b", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "z=2n-1", "name": null, "statement": "For x = 2n girls walking in rows of 2 with every pair together once, the number of days z is 2n-1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "number of days in the solution" }, { "unit": null, "symbol": "n", "meaning": "half the number of girls (x = 2n)" } ], "sympy": "Eq(z, 2*n-1)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number", "concept/solution" ] }, { "id": "ball-mathematical-recreations-1905/eq-528a861f58", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "y=35", "name": null, "statement": "For x = 15 girls, a = 3, b = 2, the greatest number of rows y is 35.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, 35)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-7d7d6df9e4", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "y=\\frac{3}{2}(x-1)/x", "name": null, "statement": "For x = 5 x 3^m girls, a = 3, b = 2, the book gives y as this expression. FLAG: as printed it gives 1.4 at x = 15, where the text states y = 35 (x(x-1)/6 gives 35); possible erratum in the book, recorded not corrected.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls, x = 5 x 3^m" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, 3*(x-1)/(2*x))", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number", "concept/solution" ] }, { "id": "ball-mathematical-recreations-1905/eq-2353d3e635", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "z=\\frac{1}{2}(x-1)", "name": null, "statement": "For x = 5 x 3^m girls, a = 3, b = 2, the number of days z is half of (x-1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls, x = 5 x 3^m" }, { "unit": null, "symbol": "z", "meaning": "number of days in the solution" } ], "sympy": "Eq(z, (x-1)/2)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-38e5936aae", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "y=x(x-1)/p(p+1)", "name": null, "statement": "If x = (p^2+p+1)(p+1) with p^2+p+1 having no divisor less than p+1, a = p+1, b = 2, then y = x(x-1)/(p(p+1)).", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "a prime-related integer in x = (p^2+p+1)(p+1)" }, { "unit": null, "symbol": "x", "meaning": "number of girls" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, x*(x-1)/(p*(p+1)))", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-97b37ee442", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "y=x", "name": null, "statement": "If x = p^3+p+1, a = p+1, b = 2, then the greatest number of ways y equals x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls, x = p^3+p+1" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, x)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-90c254313f", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "y=455", "name": null, "statement": "For x = 15, a = 3, b = 3 (Sylvester's result), the greatest number of ways y is 455.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, 455)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-9b37697379", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "z=91", "name": null, "statement": "For x = 15, a = 3, b = 3, the number of days z is 91.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "number of days in the solution" } ], "sympy": "Eq(z, 91)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number", "concept/solution" ] }, { "id": "ball-mathematical-recreations-1905/eq-80856b6001", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "y=84", "name": null, "statement": "For x = 9, a = 3, b = 3, the greatest number of ways y is 84.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, 84)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-130477cf9b", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 144", "location": "Some Miscellaneous Questions", "latex": "z=28", "name": null, "statement": "For x = 9, a = 3, b = 3, the number of days z is 28.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "number of days in the solution" } ], "sympy": "Eq(z, 28)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number", "concept/solution" ] }, { "id": "ball-mathematical-recreations-1905/eq-0d4542329f", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 145", "location": "Some Miscellaneous Questions", "latex": "y=7", "name": null, "statement": "For x = 7, a = 3, b = 2 (Bills), the greatest number of ways y is 7.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, 7)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-f2fad6b3f4", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 145", "location": "Some Miscellaneous Questions", "latex": "y=155", "name": null, "statement": "For x = 31, a = 3, b = 2 (Bills), the greatest number of ways y is 155.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, 155)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-e0619e6433", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 145", "location": "Some Miscellaneous Questions", "latex": "y=66", "name": null, "statement": "For x = 11, a = 5, b = 4 (Lea), the greatest number of ways y is 66.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, 66)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-b6223fc5e7", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 145", "location": "Some Miscellaneous Questions", "latex": "y=140", "name": null, "statement": "For x = 16, a = 4, b = 3 (Lea), the greatest number of ways y is 140.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number of girls" }, { "unit": null, "symbol": "y", "meaning": "greatest number of ways (rows) in the arrangement" } ], "sympy": "Eq(y, 140)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-24df0367f6", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 140", "location": "Some Miscellaneous Questions", "latex": "(455)^7", "name": null, "statement": "The total number of ways the school of fifteen girls can walk out in triplets for a week is (455)^7, so a chance arrangement satisfying Kirkman's condition is very improbable.", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "total number of ways of walking out in triplets for a week (not named in the text)" } ], "sympy": "Eq(N, 455**7)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/probability", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-e554d25a59", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 140", "location": "Some Miscellaneous Questions", "latex": "15567,552000", "name": null, "statement": "Power showed there are this many different solutions of the fifteen school-girls problem. FLAG: the printed number has a comma inside the digits (15567,552000); read here as 15567552000, which may be a typesetting error to check.", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "number of different solutions (Power's count)" } ], "sympy": "Eq(N, 15567552000)", "physics": false, "states": [], "concepts": [ "concept/kirkman-s-schoolgirl-problem", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-1b9074e714", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 149", "location": "Some Miscellaneous Questions", "latex": "\\frac{1}{2}n(n+1)", "name": null, "statement": "A pack of n(n+1) cards divided into couples contains one half of n(n+1) couples, which equals the number of homogeneous products of two dimensions that can be formed out of n things.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of rows in the dealing scheme (the pack is dealt into n rows of n+1 cards)" } ], "sympy": "Eq(N, n*(n+1)/2)", "physics": false, "states": [], "concepts": [ "concept/couple", "concept/homogeneous-polynomial", "concept/real-number", "method/pairs-of-cards-trick" ] }, { "id": "ball-mathematical-recreations-1905/eq-f55dd01c56", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 152", "location": "Some Miscellaneous Questions", "latex": "m^m", "name": null, "statement": "The pack size for Gergonne's generalization is m to the power m, dealt into m piles of m^(m-1) cards each.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "m", "meaning": "number of piles and number of deals" } ], "sympy": "Eq(N, m**m)", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/real-number", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-39afb562ae", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 152", "location": "Some Miscellaneous Questions", "latex": "n=km^{m-1}-jm^{m-2} + \\dotsb + bm - a + 1", "name": "Gergonne's formula", "statement": "When m is even, after m deals into m piles, the selected card is the n-th card from the top, where n is fixed by the piles a, b, ..., k taken up after each deal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "position of the selected card from the top of the collected pack" }, { "unit": null, "symbol": "m", "meaning": "number of piles and number of deals (pack of m^m cards)" }, { "unit": null, "symbol": "k", "meaning": "pile position (1 to m) in which the selected card lies, taken up last after the m-th deal" }, { "unit": null, "symbol": "j", "meaning": "pile position taken up after the (m-1)-th deal (the letter sequence continues a, b, ..., j, k)" }, { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal" }, { "unit": null, "symbol": "a", "meaning": "pile position taken up after the first deal" } ], "sympy": null, "physics": false, "states": [ "theorem/gergonne-s-theorem" ], "concepts": [ "concept/exponent", "concept/integer", "method/three-pile-problem" ] }, { "id": "ball-mathematical-recreations-1905/eq-ae3f8b6617", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 152", "location": "Some Miscellaneous Questions", "latex": "n=km^{m-1}-jm^{m-2} + \\dotsb - bm + a", "name": "Gergonne's formula", "statement": "When m is odd, the selected card is the n-th card from the top of the collected pack, with the signs of the terms alternating from the pile positions a, b, ..., k.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "position of the selected card from the top of the collected pack" }, { "unit": null, "symbol": "m", "meaning": "number of piles and number of deals" }, { "unit": null, "symbol": "k", "meaning": "pile position taken up last after the m-th deal" }, { "unit": null, "symbol": "j", "meaning": "pile position taken up after the (m-1)-th deal" }, { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal" }, { "unit": null, "symbol": "a", "meaning": "pile position taken up after the first deal" } ], "sympy": null, "physics": false, "states": [ "theorem/gergonne-s-theorem" ], "concepts": [ "concept/exponent", "concept/integer", "method/three-pile-problem" ] }, { "id": "ball-mathematical-recreations-1905/eq-c9450cf8e7", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 153", "location": "Some Miscellaneous Questions", "latex": "64d-16c + 4b-a + 1", "name": null, "statement": "In a pack of 256 cards dealt four times into four piles of 64, the selected card is the (64d-16c+4b-a+1)-th card from the top, where a, b, c, d are the pile positions taken up after each deal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "d", "meaning": "pile position taken up after the fourth deal" }, { "unit": null, "symbol": "c", "meaning": "pile position taken up after the third deal" }, { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal" }, { "unit": null, "symbol": "a", "meaning": "pile position taken up after the first deal" } ], "sympy": "Eq(n, 64*d - 16*c + 4*b - a + 1)", "physics": false, "states": [], "concepts": [ "concept/integer", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-c8fcb70de5", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 153", "location": "Some Miscellaneous Questions", "latex": "64-16c + 4b-a + 1", "name": null, "statement": "After the fourth deal of a 256-card pack, the selected card is the (64-16c+4b-a+1)-th card in the pile indicated as containing it.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "pile position taken up after the third deal" }, { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal" }, { "unit": null, "symbol": "a", "meaning": "pile position taken up after the first deal" } ], "sympy": "Eq(p, 64 - 16*c + 4*b - a + 1)", "physics": false, "states": [], "concepts": [ "concept/integer", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-a1784b9886", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 154", "location": "Some Miscellaneous Questions", "latex": "n = 9c-3b + a", "name": null, "statement": "In a 27-card pack dealt three times into three piles of nine, the selected card is the n-th card from the top when the piles taken up are a, b, c.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "position of the selected card from the top of the pack" }, { "unit": null, "symbol": "a", "meaning": "pile position taken up after the first deal" }, { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal" }, { "unit": null, "symbol": "c", "meaning": "pile position taken up after the third deal" } ], "sympy": "Eq(n, 9*c - 3*b + a)", "physics": false, "states": [], "concepts": [ "concept/integer", "method/three-pile-problem", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-2e8aaa9ca4", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 153", "location": "Some Miscellaneous Questions", "latex": "9-3b + a", "name": null, "statement": "After the third deal, the selected card is the (9-3b+a)-th card from the top of the pile indicated as containing it.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal" }, { "unit": null, "symbol": "a", "meaning": "pile position taken up after the first deal" } ], "sympy": "Eq(q, 9 - 3*b + a)", "physics": false, "states": [], "concepts": [ "concept/integer", "method/three-pile-problem", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-136f91922a", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 154", "location": "Some Miscellaneous Questions", "latex": "9(c-1) + (8-3b + a) + 1", "name": null, "statement": "The selected card lies in the place 9(c-1)+(8-3b+a)+1 from the top of the pack when the pile indicated after the third deal is taken up c-th.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "pile position taken up after the third deal" }, { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal" }, { "unit": null, "symbol": "a", "meaning": "pile position taken up after the first deal" } ], "sympy": "Eq(p, 9*(c - 1) + (8 - 3*b + a) + 1)", "physics": false, "states": [], "concepts": [ "concept/integer", "method/three-pile-problem", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-a1166c63be", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 154", "location": "Some Miscellaneous Questions", "latex": "n = 9c-3b + a= 14", "name": null, "statement": "With the pile indicated always taken up in the middle (a = b = c = 2), the selected card is the 14th card from the top, as in the usual presentation of the trick.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "position of the selected card from the top of the 27-card pack" }, { "unit": null, "symbol": "a", "meaning": "pile position taken up after the first deal (here 2, the middle pile)" }, { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal (here 2)" }, { "unit": null, "symbol": "c", "meaning": "pile position taken up after the third deal (here 2)" } ], "sympy": "Eq(9*c - 3*b + a, 14)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/middle", "method/three-pile-problem", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-55ced50757", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 155", "location": "Some Miscellaneous Questions", "latex": "3p = 9c-3b", "name": null, "statement": "Dividing the adjusted position by three gives a multiple of three equal to 9c minus 3b, which is used to solve for the pile positions.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "the multiple of 3 next lowest to n/3" }, { "unit": null, "symbol": "c", "meaning": "pile position taken up after the third deal" }, { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal" } ], "sympy": "Eq(3*p, 9*c - 3*b)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/multiple", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-eff498fcee", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 155", "location": "Some Miscellaneous Questions", "latex": "p = 3c-b", "name": null, "statement": "The integer p equals 3c minus b, so b is the smallest positive number that added to p gives a multiple of three, and c is that multiple.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "the next lowest integer to n/3" }, { "unit": null, "symbol": "c", "meaning": "pile position taken up after the third deal" }, { "unit": null, "symbol": "b", "meaning": "pile position taken up after the second deal" } ], "sympy": "Eq(p, 3*c - b)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/multiple", "concept/remainder", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-b4377910ed", "chapter": "ball-mathematical-recreations-1905/ch-iv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 155", "location": "Some Miscellaneous Questions", "latex": "9z + 3y + x = n-1", "name": "Gergonne's equation", "statement": "Gergonne's equation rewritten with x = a-1, y = 3-b, z = c-1, so that n-1 expressed in the ternary scale gives x, y, z.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "c minus 1, a digit of the ternary expansion of n-1 (values 0, 1, 2)" }, { "unit": null, "symbol": "y", "meaning": "3 minus b, a digit of the ternary expansion of n-1 (values 0, 1, 2)" }, { "unit": null, "symbol": "x", "meaning": "a minus 1, a digit of the ternary expansion of n-1 (values 0, 1, 2)" }, { "unit": null, "symbol": "n", "meaning": "position of the selected card from the top of the 27-card pack" } ], "sympy": "Eq(9*z + 3*y + x, n - 1)", "physics": false, "states": [ "theorem/gergonne-s-equation" ], "concepts": [ "concept/integer", "concept/remainder", "concept/ternary-scale-of-notation", "theorem/gergonne-s-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-4aa5141752", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 167", "location": "Magic Squares", "latex": "N = \\frac{1}{2}n (n^2 + 1)", "name": null, "statement": "The magic sum N, the common total of every row, column and diagonal of a magic square of order n filled with the integers 1 to n^2, equals one half of n times (n^2 + 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "the magic sum: the common sum of the numbers in each row, column and diagonal" }, { "unit": null, "symbol": "n", "meaning": "the order of the magic square (the square has n^2 cells)" } ], "sympy": "Eq(N, n*(n**2 + 1)/2)", "physics": false, "states": [], "concepts": [ "concept/magic-square", "concept/real-number", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-92583284c1", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 162", "location": "Magic Squares", "latex": "\\frac{1}{2}n(n-1)", "name": null, "statement": "Each of the n radix-digits in a diagonal of the square built by De la Hire's method has the value one half of n(n-1), so the diagonal sums to that multiple of n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "the order of the magic square" } ], "sympy": "Eq(d, n*(n - 1)/2)", "physics": false, "states": [], "concepts": [ "concept/denary-scale-of-notation", "concept/diagonal", "concept/digit", "concept/odd-magic-square", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-4df55d172a", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 162", "location": "Magic Squares", "latex": "\\frac{1}{2}(n+1)", "name": null, "statement": "Each of the n unit-digits in a diagonal of an odd magic square built by De la Loubère's method equals one half of (n + 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "the order of the magic square" } ], "sympy": "Eq(u, (n + 1)/2)", "physics": false, "states": [], "concepts": [ "concept/diagonal", "concept/digit", "concept/odd-magic-square", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-8dab8c6118", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 168", "location": "Magic Squares", "latex": "n(n-2x + 1)", "name": null, "statement": "In the x-th row from the top, the number in a cell is less than the number in the vertically related cell in the complementary row by n(n - 2x + 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "the order of the magic square" }, { "unit": null, "symbol": "x", "meaning": "the index of a row counted from the top" } ], "sympy": "Eq(D, n*(n - 2*x + 1))", "physics": false, "states": [], "concepts": [ "concept/complementary-rows-and-columns", "concept/element-of-a-determinant", "concept/magic-square", "concept/vertically-related-cells" ] }, { "id": "ball-mathematical-recreations-1905/eq-d1ad0234a4", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 168", "location": "Magic Squares", "latex": "n-2y+1", "name": null, "statement": "In the y-th column from the left, the number in a cell is less than the number in the horizontally related cell in the complementary column by n - 2y + 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "the order of the magic square" }, { "unit": null, "symbol": "y", "meaning": "the index of a column counted from the left" } ], "sympy": "Eq(D_h, n - 2*y + 1)", "physics": false, "states": [], "concepts": [ "concept/complementary-rows-and-columns", "concept/horizontally-related-cells", "concept/magic-square" ] }, { "id": "ball-mathematical-recreations-1905/eq-029f4b08dd", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 168", "location": "Magic Squares", "latex": "N-\\frac{1}{2}n^2(n-2x+1)", "name": null, "statement": "The sum of the numbers in the x-th row from the top of a square of order n, before interchanges, is N minus one half of n^2 times (n - 2x + 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "the magic sum" }, { "unit": null, "symbol": "n", "meaning": "the order of the square" }, { "unit": null, "symbol": "x", "meaning": "the index of the row counted from the top" }, { "unit": null, "symbol": "S", "meaning": "the sum of the numbers in the x-th row from the top" } ], "sympy": "Eq(S, N - n**2*(n - 2*x + 1)/2)", "physics": false, "states": [], "concepts": [ "concept/complementary-rows-and-columns", "concept/element-of-a-determinant", "concept/magic-square-of-an-even-order", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-f1a81f0618", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 168", "location": "Magic Squares", "latex": "N + \\frac{1}{2}n^2(n-2x + 1)", "name": null, "statement": "The sum of the numbers in the complementary row, the x-th row from the bottom, before interchanges, is N plus one half of n^2 times (n - 2x + 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "the magic sum" }, { "unit": null, "symbol": "n", "meaning": "the order of the square" }, { "unit": null, "symbol": "x", "meaning": "the index of the row counted from the top of the pair" }, { "unit": null, "symbol": "S", "meaning": "the sum of the numbers in the complementary row (the x-th row from the bottom)" } ], "sympy": "Eq(S, N + n**2*(n - 2*x + 1)/2)", "physics": false, "states": [], "concepts": [ "concept/complementary-rows-and-columns", "concept/element-of-a-determinant", "concept/magic-square-of-an-even-order", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-8a8eda536a", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 168", "location": "Magic Squares", "latex": "N-\\frac{1}{2}n (n-2y + 1)", "name": null, "statement": "The sum of the numbers originally in the y-th column from the left, of a square of order n in natural order, is N minus one half of n times (n - 2y + 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "the magic sum" }, { "unit": null, "symbol": "n", "meaning": "the order of the square" }, { "unit": null, "symbol": "y", "meaning": "the index of the column counted from the left" }, { "unit": null, "symbol": "S", "meaning": "the sum of the numbers originally in the y-th column from the left" } ], "sympy": "Eq(S, N - n*(n - 2*y + 1)/2)", "physics": false, "states": [], "concepts": [ "concept/complementary-rows-and-columns", "concept/element-of-a-determinant", "concept/magic-square-of-an-even-order", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-aa6c93d3b5", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 168", "location": "Magic Squares", "latex": "N + \\frac{1}{2}n(n-2y +1)", "name": null, "statement": "The sum of the numbers originally in the complementary column, the y-th column from the right, is N plus one half of n times (n - 2y + 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "N", "meaning": "the magic sum" }, { "unit": null, "symbol": "n", "meaning": "the order of the square" }, { "unit": null, "symbol": "y", "meaning": "the index of the column counted from the left" }, { "unit": null, "symbol": "S", "meaning": "the sum of the numbers originally in the complementary column (the y-th column from the right)" } ], "sympy": "Eq(S, N + n*(n - 2*y + 1)/2)", "physics": false, "states": [], "concepts": [ "concept/complementary-rows-and-columns", "concept/element-of-a-determinant", "concept/magic-square-of-an-even-order", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-dcb09d0eb7", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 175", "location": "Magic Squares", "latex": "\\frac{1}{2}(n^2 + 1)", "name": null, "statement": "The average number in a magic square of the nth order is one half of n squared plus one.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "order of the magic square (number of cells along one side)" }, { "unit": null, "symbol": "A", "meaning": "average number in a magic square of order n" } ], "sympy": "Eq(A, (n**2 + 1)/2)", "physics": false, "states": [], "concepts": [ "concept/average", "concept/magic-square", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-1b5b81af19", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 175", "location": "Magic Squares", "latex": "\\frac{1}{2}(n-2)\\{(n-2)^2+1\\}", "name": null, "statement": "The sum of the numbers in each line of a magic square of order n-2 (the inner square of a bordered square) is one half of (n-2) times ((n-2) squared plus one).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "order of the bordered square" }, { "unit": null, "symbol": "S", "meaning": "sum of the numbers in each line of the inner square of order n-2" } ], "sympy": "Eq(S, (n-2)*((n-2)**2 + 1)/2)", "physics": false, "states": [], "concepts": [ "concept/bordered-magic-square", "concept/line", "concept/magic-square", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-c30e82734d", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 175", "location": "Magic Squares", "latex": "\\frac{1}{2}\\{(n-2)^2+1\\}", "name": null, "statement": "The average number in a magic square of order n-2 is one half of ((n-2) squared plus one).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "order of the bordered square" }, { "unit": null, "symbol": "a", "meaning": "average number in the inner magic square of order n-2" } ], "sympy": "Eq(a, ((n-2)**2 + 1)/2)", "physics": false, "states": [], "concepts": [ "concept/average", "concept/bordered-magic-square", "concept/magic-square", "concept/sum" ] }, { "id": "ball-mathematical-recreations-1905/eq-84d3c61198", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 175", "location": "Magic Squares", "latex": "2(n-1)", "name": null, "statement": "The difference between the average number of an nth-order square and that of an (n-2)th-order square is two times (n-1), so every number of the inner square is raised by this amount.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "order of the bordered square" }, { "unit": null, "symbol": "D", "meaning": "difference between the two averages" } ], "sympy": "Eq(D, 2*(n-1))", "physics": false, "states": [], "concepts": [ "concept/average", "concept/bordered-magic-square", "concept/difference" ] }, { "id": "ball-mathematical-recreations-1905/eq-79bc11cc12", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 175", "location": "Magic Squares", "latex": "n^2 + 1-p", "name": null, "statement": "The number that must be placed opposite the number p in a bordered magic square of n squared cells is n squared plus one minus p; the book denotes it by p with a bar.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "order of the square (it holds n squared cells)" }, { "unit": null, "symbol": "p", "meaning": "a number placed in the border" }, { "unit": null, "symbol": "\\overline{p}", "meaning": "the number opposite to p, equal to n squared plus one minus p" } ], "sympy": "Eq(pbar, n**2 + 1 - p)", "physics": false, "states": [], "concepts": [ "concept/bordered-magic-square", "concept/complementary-numbers", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-52a34e7518", "chapter": "ball-mathematical-recreations-1905/ch-v", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 182", "location": "Magic Squares", "latex": "\\frac{1}{2}(n+1)(n+2)", "name": null, "statement": "A double-domino set running from double zero to double n contains one half of (n+1)(n+2) dominoes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "the highest pip count on the double dominoes (set ranges from double zero to double n)" }, { "unit": null, "symbol": "D", "meaning": "number of dominoes in the set" } ], "sympy": "Eq(D, (n+1)*(n+2)/2)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-06b0159307", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 198", "location": "Unicursal Problems", "latex": "n^{n-2}", "name": null, "statement": "The number of trees with n given nodes is n raised to the power n-2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of given nodes of the tree" } ], "sympy": "Eq(T, n**(n-2))", "physics": false, "states": [], "concepts": [ "concept/node", "concept/real-number", "concept/tree" ] }, { "id": "ball-mathematical-recreations-1905/eq-872dc18479", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 198", "location": "Unicursal Problems", "latex": "(1-x)^{-1} (1-x^2)^{-A_1} (1-x^3)^{-A_2} \\dotsm & = 1 + A_1 x + A_2 x^2 + A_3 x^3 + \\dotsb\\, ,", "name": null, "statement": "A generating-function identity: an infinite product in x equals the series whose coefficients are the numbers A_n of trees with n branches.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "formal variable of the generating series" }, { "unit": null, "symbol": "A_n", "meaning": "number of trees with n branches" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/real-number", "concept/tree" ] }, { "id": "ball-mathematical-recreations-1905/eq-3b06e11cd2", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 198", "location": "Unicursal Problems", "latex": "(1-x)^{-1} (1-x^2)^{-B_2} (1-x^3)^{-B_3} \\dotsm & = 1 + x + 2B_2 x^2 + 2B_3 x^3 + \\dotsb\\,.", "name": null, "statement": "A generating-function identity: an infinite product in x equals the series whose coefficients involve the numbers B_n of trees with n free branches that are bifurcations at least.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "formal variable of the generating series" }, { "unit": null, "symbol": "B_n", "meaning": "number of trees with n free branches which are bifurcations at least" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/real-number", "concept/tree" ] }, { "id": "ball-mathematical-recreations-1905/eq-33996dabdb", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 200", "location": "Unicursal Problems", "latex": "lr^2l = rlr", "name": null, "statement": "Making the move left, then two right moves, then left has the same total effect as right, left, right on a dodecahedron's edges.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "l", "meaning": "operation: take the edge to the left at each angular point" }, { "unit": null, "symbol": "r", "meaning": "operation: take the edge to the right at each angular point" } ], "sympy": "Eq(l*r**2*l, r*l*r)", "physics": false, "states": [], "concepts": [ "concept/hamiltonian-cycle", "concept/left-right-operation-notation", "concept/node", "concept/operation" ] }, { "id": "ball-mathematical-recreations-1905/eq-21de75d737", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 200", "location": "Unicursal Problems", "latex": "rl^2r = lrl", "name": null, "statement": "Making the move right, then two left moves, then right has the same total effect as left, right, left on a dodecahedron's edges.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "l", "meaning": "operation: take the edge to the left at each angular point" }, { "unit": null, "symbol": "r", "meaning": "operation: take the edge to the right at each angular point" } ], "sympy": "Eq(r*l**2*r, l*r*l)", "physics": false, "states": [], "concepts": [ "concept/hamiltonian-cycle", "concept/left-right-operation-notation", "concept/node", "concept/operation" ] }, { "id": "ball-mathematical-recreations-1905/eq-b5119ee40d", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 200", "location": "Unicursal Problems", "latex": "lr^3l=r^2", "name": null, "statement": "Making the move left, three right moves, then left has the same total effect as two right moves.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "l", "meaning": "operation: take the edge to the left at each angular point" }, { "unit": null, "symbol": "r", "meaning": "operation: take the edge to the right at each angular point" } ], "sympy": "Eq(l*r**3*l, r**2)", "physics": false, "states": [], "concepts": [ "concept/hamiltonian-cycle", "concept/left-right-operation-notation", "concept/node", "concept/operation" ] }, { "id": "ball-mathematical-recreations-1905/eq-9e020c4c96", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 200", "location": "Unicursal Problems", "latex": "rl^3r=l^2", "name": null, "statement": "Making the move right, three left moves, then right has the same total effect as two left moves.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "l", "meaning": "operation: take the edge to the left at each angular point" }, { "unit": null, "symbol": "r", "meaning": "operation: take the edge to the right at each angular point" } ], "sympy": "Eq(r*l**3*r, l**2)", "physics": false, "states": [], "concepts": [ "concept/hamiltonian-cycle", "concept/left-right-operation-notation", "concept/node", "concept/operation" ] }, { "id": "ball-mathematical-recreations-1905/eq-8f858dc55b", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 200", "location": "Unicursal Problems", "latex": "l^5=1", "name": null, "statement": "Five successive left moves bring the traveller back to the starting point.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "l", "meaning": "operation: take the edge to the left at each angular point" } ], "sympy": "Eq(l**5, 1)", "physics": false, "states": [], "concepts": [ "concept/hamiltonian-cycle", "concept/left-right-operation-notation", "concept/operation" ] }, { "id": "ball-mathematical-recreations-1905/eq-a6ff5d8f22", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 200", "location": "Unicursal Problems", "latex": "r^5=1", "name": null, "statement": "Five successive right moves bring the traveller back to the starting point.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "r", "meaning": "operation: take the edge to the right at each angular point" } ], "sympy": "Eq(r**5, 1)", "physics": false, "states": [], "concepts": [ "concept/hamiltonian-cycle", "concept/left-right-operation-notation", "concept/operation" ] }, { "id": "ball-mathematical-recreations-1905/eq-96fe6230f9", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 200", "location": "Unicursal Problems", "latex": "\\{r^3l^3(rl)^2\\}^2=1", "name": null, "statement": "The twenty-step operation r r r l l l r l r l r r r l l l r l r l, read cyclically, returns to the start, so it traces a route through every town on the dodecahedron (condition (i)).", "kind": "result", "symbols": [ { "unit": null, "symbol": "l", "meaning": "operation: take the edge to the left at each angular point" }, { "unit": null, "symbol": "r", "meaning": "operation: take the edge to the right at each angular point" } ], "sympy": "Eq((r**3*l**3*(r*l)**2)**2, 1)", "physics": false, "states": [], "concepts": [ "concept/hamiltonian-cycle", "concept/left-right-operation-notation", "concept/operation", "concept/route" ] }, { "id": "ball-mathematical-recreations-1905/eq-d89825597f", "chapter": "ball-mathematical-recreations-1905/ch-vi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 200", "location": "Unicursal Problems", "latex": "\\{l^3r^3(lr)^2\\}^2=1", "name": null, "statement": "The twenty-step operation l l l r r r l r l r l l l r r r l r l r, read cyclically, returns to the start, so it traces a route through every town on the dodecahedron (condition (ii)).", "kind": "result", "symbols": [ { "unit": null, "symbol": "l", "meaning": "operation: take the edge to the left at each angular point" }, { "unit": null, "symbol": "r", "meaning": "operation: take the edge to the right at each angular point" } ], "sympy": "Eq((l**3*r**3*(l*r)**2)**2, 1)", "physics": false, "states": [], "concepts": [ "concept/hamiltonian-cycle", "concept/left-right-operation-notation", "concept/operation", "concept/route" ] }, { "id": "ball-mathematical-recreations-1905/eq-30020875e1", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 261", "location": "Three Geometrical Problems", "latex": "a : x = x : y = y : b", "name": null, "statement": "Descartes's intersection of the curves gives x and y as the two mean proportionals between a and b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "given line" }, { "unit": null, "symbol": "b", "meaning": "second given line" }, { "unit": null, "symbol": "x", "meaning": "first mean proportional" }, { "unit": null, "symbol": "y", "meaning": "second mean proportional" } ], "sympy": "Eq(a/x, x/y)", "physics": false, "states": [], "concepts": [ "concept/means-of-a-proportion", "concept/proportion" ] }, { "id": "ball-mathematical-recreations-1905/eq-a82c0c42c2", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 255", "location": "Three Geometrical Problems", "latex": "x^3 = 2a^3", "name": "duplication of the cube condition", "statement": "The side x of the required cube, whose volume is double that of a cube of side a, satisfies x cubed equals twice a cubed.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "side of the required cube" }, { "unit": null, "symbol": "a", "meaning": "side of the given cube" } ], "sympy": "Eq(x**3, 2*a**3)", "physics": false, "states": [ "theorem/duplication-of-the-cube-condition" ], "concepts": [ "concept/cubic-equation", "concept/duplication-of-the-cube", "concept/hexahedron" ] }, { "id": "ball-mathematical-recreations-1905/eq-4c54602da6", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 255", "location": "Three Geometrical Problems", "latex": "4x^3=3x-a", "name": null, "statement": "If x is the sine of an angle one-third of a given angle whose sine is a, then x satisfies this cubic equation.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "sine of one-third of the angle" }, { "unit": null, "symbol": "a", "meaning": "sine of the given angle" } ], "sympy": "Eq(4*x**3, 3*x - a)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/sine", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-43b9e85cc4", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 255", "location": "Three Geometrical Problems", "latex": "x^2 + y^2 + ax + by + c = 0", "name": null, "statement": "The general form of the equation of a circle in Cartesian coordinates.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of a point" }, { "unit": null, "symbol": "y", "meaning": "ordinate of a point" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x" }, { "unit": null, "symbol": "b", "meaning": "coefficient of y" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq(x**2 + y**2 + a*x + b*y + c, 0)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/circle", "concept/conic-section" ] }, { "id": "ball-mathematical-recreations-1905/eq-deb71e7d9f", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 255", "location": "Three Geometrical Problems", "latex": "\\alpha x + \\beta y +\\gamma = 0", "name": null, "statement": "The general form of the equation of a straight line in Cartesian coordinates.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of a point" }, { "unit": null, "symbol": "y", "meaning": "ordinate of a point" }, { "unit": null, "symbol": "\\alpha", "meaning": "coefficient of x" }, { "unit": null, "symbol": "\\beta", "meaning": "coefficient of y" }, { "unit": null, "symbol": "\\gamma", "meaning": "constant term" } ], "sympy": "Eq(alpha*x + beta*y + gamma, 0)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/line" ] }, { "id": "ball-mathematical-recreations-1905/eq-8c22cd2f71", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 257", "location": "Three Geometrical Problems", "latex": "x: a = \\sqrt[3]{2}: 1", "name": null, "statement": "The ratio of the side of the required cube to the side of the given cube is the cube root of two.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "side of the required cube" }, { "unit": null, "symbol": "a", "meaning": "side of the given cube" } ], "sympy": "Eq(x/a, 2**Rational(1, 3))", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/duplication-of-the-cube", "concept/incommensurable-magnitudes", "concept/root" ] }, { "id": "ball-mathematical-recreations-1905/eq-25cb39eb5f", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 258", "location": "Three Geometrical Problems", "latex": "a: x = x: y = y: 2a", "name": null, "statement": "Two mean proportionals x and y lie between a and 2a, so that the problem of the duplicated cube is that of finding two means between a line and one twice as long.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "the given straight line" }, { "unit": null, "symbol": "x", "meaning": "first mean proportional" }, { "unit": null, "symbol": "y", "meaning": "second mean proportional" } ], "sympy": "Eq(a/x, x/y)", "physics": false, "states": [], "concepts": [ "concept/duplication-of-the-cube", "concept/means-of-a-proportion", "concept/proportion" ] }, { "id": "ball-mathematical-recreations-1905/eq-415022ec5c", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 259", "location": "Three Geometrical Problems", "latex": "r = 2a \\sin\\theta", "name": null, "statement": "In polar coordinates, the equation of the surface traced by Archytas's semicircle.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "polar radial distance" }, { "unit": null, "symbol": "\\theta", "meaning": "polar angle" }, { "unit": null, "symbol": "a", "meaning": "radius of the cylinder" } ], "sympy": "Eq(r, 2*a*sin(theta))", "physics": false, "states": [], "concepts": [ "concept/cylindrical-surface", "concept/duplication-of-the-cube", "concept/polar-coordinates", "concept/sine" ] }, { "id": "ball-mathematical-recreations-1905/eq-aa240f72ea", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 259", "location": "Three Geometrical Problems", "latex": "r \\sin\\theta = 2a \\cos\\phi", "name": null, "statement": "The equation of the cylinder used in Archytas's construction, in polar coordinates.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "polar radial distance" }, { "unit": null, "symbol": "\\theta", "meaning": "polar angle" }, { "unit": null, "symbol": "\\phi", "meaning": "angle in the base plane" }, { "unit": null, "symbol": "a", "meaning": "radius of the cylinder" } ], "sympy": "Eq(r*sin(theta), 2*a*cos(phi))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cylindrical-surface", "concept/polar-coordinates" ] }, { "id": "ball-mathematical-recreations-1905/eq-9bde780238", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 259", "location": "Three Geometrical Problems", "latex": "\\sin\\theta \\cos\\phi = \\frac{1}{2}", "name": null, "statement": "The equation of the right cone used in Archytas's construction.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "\\theta", "meaning": "polar angle" }, { "unit": null, "symbol": "\\phi", "meaning": "angle in the base plane" } ], "sympy": "Eq(sin(theta)*cos(phi), 1/2)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/right-circular-cone", "concept/sine" ] }, { "id": "ball-mathematical-recreations-1905/eq-4e4d29a1d9", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 259", "location": "Three Geometrical Problems", "latex": "\\sin^3\\theta = \\frac{1}{2}", "name": null, "statement": "At the point where the three surfaces meet, sin cubed theta equals one half.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\theta", "meaning": "polar angle of the intersection point" } ], "sympy": "Eq(sin(theta)**3, Rational(1, 2))", "physics": false, "states": [], "concepts": [ "concept/duplication-of-the-cube", "concept/hexahedron", "concept/sine" ] }, { "id": "ball-mathematical-recreations-1905/eq-63c24ab59e", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 259", "location": "Three Geometrical Problems", "latex": "(r\\sin\\theta)^3=2a^3", "name": null, "statement": "The cube on side r sin theta has twice the volume of the cube on side a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "polar radial distance of the intersection point" }, { "unit": null, "symbol": "\\theta", "meaning": "polar angle" }, { "unit": null, "symbol": "a", "meaning": "radius of the cylinder" } ], "sympy": "Eq((r*sin(theta))**3, 2*a**3)", "physics": false, "states": [], "concepts": [ "concept/duplication-of-the-cube", "concept/hexahedron", "quantity/volume" ] }, { "id": "ball-mathematical-recreations-1905/eq-d31d808c30", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 259", "location": "Three Geometrical Problems", "latex": "PC : PB = PB : PA = PA : PD", "name": null, "statement": "In Plato's figure, the segments PC, PB, PA and PD are in continued proportion.", "kind": "result", "symbols": [ { "unit": null, "symbol": "PC", "meaning": "segment of the hypotenuse of the first triangle" }, { "unit": null, "symbol": "PB", "meaning": "segment from P to B" }, { "unit": null, "symbol": "PA", "meaning": "segment from P to A" }, { "unit": null, "symbol": "PD", "meaning": "segment from P to D" } ], "sympy": "Eq(PC/PB, PB/PA)", "physics": false, "states": [], "concepts": [ "concept/duplication-of-the-cube", "concept/proportion", "concept/right-triangle" ] }, { "id": "ball-mathematical-recreations-1905/eq-d2e13dbfaa", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 259", "location": "Three Geometrical Problems", "latex": "y^2 = 2ax", "name": null, "statement": "Equation of the first parabola in Menaechmus's first solution, with latus rectum double that of the second.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "a", "meaning": "parameter of the parabola" } ], "sympy": "Eq(y**2, 2*a*x)", "physics": false, "states": [], "concepts": [ "concept/latus-rectum", "concept/parabola", "concept/vertex-of-a-conic-section" ] }, { "id": "ball-mathematical-recreations-1905/eq-0085ee58ba", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 259", "location": "Three Geometrical Problems", "latex": "x^2=ay", "name": null, "statement": "Equation of the second parabola in Menaechmus's first solution.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "a", "meaning": "parameter of the parabola" } ], "sympy": "Eq(x**2, a*y)", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/parabola" ] }, { "id": "ball-mathematical-recreations-1905/eq-165535994a", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 260", "location": "Three Geometrical Problems", "latex": "x^3 = 2l^3", "name": null, "statement": "The abscissa of the intersection of Menaechmus's parabola and rectangular hyperbola satisfies x cubed equals twice l cubed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of the intersection point" }, { "unit": null, "symbol": "l", "meaning": "latus rectum of the parabola" } ], "sympy": "Eq(x**3, 2*l**3)", "physics": false, "states": [], "concepts": [ "concept/duplication-of-the-cube", "concept/hexahedron", "concept/hyperbola", "concept/parabola" ] }, { "id": "ball-mathematical-recreations-1905/eq-591644ece6", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 260", "location": "Three Geometrical Problems", "latex": "y^3 = 4l^3", "name": null, "statement": "The ordinate of the intersection of Menaechmus's curves satisfies y cubed equals four l cubed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate of the intersection point" }, { "unit": null, "symbol": "l", "meaning": "latus rectum of the parabola" } ], "sympy": "Eq(y**3, 4*l**3)", "physics": false, "states": [], "concepts": [ "concept/hexahedron", "concept/hyperbola", "concept/parabola" ] }, { "id": "ball-mathematical-recreations-1905/eq-894d31e6df", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 260", "location": "Three Geometrical Problems", "latex": "x^2 = ly", "name": null, "statement": "Equation of the parabola of latus rectum l in Menaechmus's second solution.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "l", "meaning": "latus rectum" } ], "sympy": "Eq(x**2, l*y)", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/latus-rectum", "concept/parabola" ] }, { "id": "ball-mathematical-recreations-1905/eq-e87187e2c1", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 260", "location": "Three Geometrical Problems", "latex": "xy = 2l^2", "name": null, "statement": "Equation of the rectangular hyperbola in Menaechmus's second solution.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "l", "meaning": "latus rectum of the parabola" } ], "sympy": "Eq(x*y, 2*l**2)", "physics": false, "states": [], "concepts": [ "concept/asymptote", "concept/conic-section", "concept/hyperbola" ] }, { "id": "ball-mathematical-recreations-1905/eq-eade4e93a0", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 260", "location": "Three Geometrical Problems", "latex": "l : x = x : y = y : 2l", "name": null, "statement": "The abscissa and ordinate of the intersection are the mean proportionals between l and 2l.", "kind": "result", "symbols": [ { "unit": null, "symbol": "l", "meaning": "latus rectum of the parabola" }, { "unit": null, "symbol": "x", "meaning": "abscissa of the intersection point" }, { "unit": null, "symbol": "y", "meaning": "ordinate of the intersection point" } ], "sympy": "Eq(l/x, x/y)", "physics": false, "states": [], "concepts": [ "concept/means-of-a-proportion", "concept/proportion" ] }, { "id": "ball-mathematical-recreations-1905/eq-e185414129", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 260", "location": "Three Geometrical Problems", "latex": "OA : Bb = Bb : Aa = Aa : OB", "name": null, "statement": "Apollonius's construction gives Bb and Aa as the two mean proportionals between OA and OB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "OA", "meaning": "one of the given lines" }, { "unit": null, "symbol": "OB", "meaning": "the other given line" }, { "unit": null, "symbol": "Bb", "meaning": "first mean proportional" }, { "unit": null, "symbol": "Aa", "meaning": "second mean proportional" } ], "sympy": "Eq(OA/Bb, Bb/Aa)", "physics": false, "states": [], "concepts": [ "concept/means-of-a-proportion", "concept/proportion", "concept/rectangle" ] }, { "id": "ball-mathematical-recreations-1905/eq-df5830cdf9", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 261", "location": "Three Geometrical Problems", "latex": "x^2 + y^2 = ay + bx", "name": null, "statement": "Equation of the circle used in Descartes's and Gregory's constructions.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "a", "meaning": "constant of the construction" }, { "unit": null, "symbol": "b", "meaning": "constant of the construction" } ], "sympy": "Eq(x**2 + y**2, a*y + b*x)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/circle" ] }, { "id": "ball-mathematical-recreations-1905/eq-3113b2ca4c", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 261", "location": "Three Geometrical Problems", "latex": "xy = ab", "name": null, "statement": "Equation of the hyperbola with the two sides of the rectangle as asymptotes, in Gregory's construction.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "distance from one asymptote" }, { "unit": null, "symbol": "y", "meaning": "distance from the other asymptote" }, { "unit": null, "symbol": "a", "meaning": "adjacent side of the rectangle" }, { "unit": null, "symbol": "b", "meaning": "adjacent side of the rectangle" } ], "sympy": "Eq(x*y, a*b)", "physics": false, "states": [], "concepts": [ "concept/asymptote", "concept/hyperbola" ] }, { "id": "ball-mathematical-recreations-1905/eq-fe03761827", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 262", "location": "Three Geometrical Problems", "latex": "BC : OD = OD : CE = CE : OA", "name": null, "statement": "In Newton's construction, OD and CE are two mean proportionals between BC and OA.", "kind": "result", "symbols": [ { "unit": null, "symbol": "BC", "meaning": "one of the given lines" }, { "unit": null, "symbol": "OA", "meaning": "the other given line" }, { "unit": null, "symbol": "OD", "meaning": "first mean proportional" }, { "unit": null, "symbol": "CE", "meaning": "second mean proportional" } ], "sympy": "Eq(BC/OD, OD/CE)", "physics": false, "states": [], "concepts": [ "concept/means-of-a-proportion", "concept/proportion" ] }, { "id": "ball-mathematical-recreations-1905/eq-7838adc7db", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 261", "location": "Three Geometrical Problems", "latex": "AB : GC = GC : GA = GA : CD", "name": null, "statement": "In Vieta's construction, GC and GA are the two mean proportionals between AB and CD.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "smaller given line" }, { "unit": null, "symbol": "CD", "meaning": "larger given line" }, { "unit": null, "symbol": "GC", "meaning": "first mean proportional" }, { "unit": null, "symbol": "GA", "meaning": "second mean proportional" } ], "sympy": "Eq(AB/GC, GC/GA)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/means-of-a-proportion", "concept/proportion" ] }, { "id": "ball-mathematical-recreations-1905/eq-01c528dab8", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 262", "location": "Three Geometrical Problems", "latex": "QR = 2\\dotm OP", "name": null, "statement": "The condition on the construction for trisecting an angle by Pappus's first method: QR is twice OP.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "QR", "meaning": "segment of the construction line" }, { "unit": null, "symbol": "OP", "meaning": "segment from O to P" } ], "sympy": "Eq(QR, 2*OP)", "physics": false, "states": [], "concepts": [ "concept/construction", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-69a51df3ae", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 262", "location": "Three Geometrical Problems", "latex": "AOR=\\frac{1}{3}AOB", "name": null, "statement": "If the construction can be made, the angle AOR is one third of the given angle AOB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AOR", "meaning": "angle to be obtained" }, { "unit": null, "symbol": "AOB", "meaning": "the given angle" } ], "sympy": "Eq(AOR, AOB/3)", "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-6858febed1", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 262", "location": "Three Geometrical Problems", "latex": "\\tan^{-1} (b/x)=\\frac{1}{3}\\tan^{-1}(b/a)", "name": null, "statement": "The angle with tangent b/x is one third of the angle with tangent b/a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "abscissa defining the given angle" }, { "unit": null, "symbol": "b", "meaning": "ordinate defining the given angle" }, { "unit": null, "symbol": "x", "meaning": "abscissa of the greatest intersection point" } ], "sympy": "Eq(atan(b/x), atan(b/a)/3)", "physics": false, "states": [], "concepts": [ "concept/arc-of-a-curve", "concept/tangent-function", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-73317d3d10", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 262", "location": "Three Geometrical Problems", "latex": "(x-a)^2 + (y-b)^2 = 4(a^2 + b^2)", "name": null, "statement": "Equation of the circle in the analytic form of Pappus's first trisection.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "a", "meaning": "abscissa of the centre" }, { "unit": null, "symbol": "b", "meaning": "ordinate of the centre" } ], "sympy": "Eq((x-a)**2 + (y-b)**2, 4*(a**2 + b**2))", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/circle" ] }, { "id": "ball-mathematical-recreations-1905/eq-597fd3eac3", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 262", "location": "Three Geometrical Problems", "latex": "PR = x-a", "name": null, "statement": "The length PR equals the abscissa x of the chosen intersection point minus a.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "PR", "meaning": "length of the segment PR" }, { "unit": null, "symbol": "x", "meaning": "greatest abscissa of intersection" }, { "unit": null, "symbol": "a", "meaning": "abscissa of the given angle" } ], "sympy": "Eq(PR, x - a)", "physics": false, "states": [], "concepts": [ "concept/abscissa", "concept/line-segment" ] }, { "id": "ball-mathematical-recreations-1905/eq-15165553aa", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 263", "location": "Three Geometrical Problems", "latex": "AOE = \\frac{1}{3} AOB", "name": null, "statement": "If the construction can be made, the angle AOE is one third of the given angle AOB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AOE", "meaning": "angle to be obtained" }, { "unit": null, "symbol": "AOB", "meaning": "the given angle" } ], "sympy": "Eq(AOE, AOB/3)", "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-79897189e9", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 263", "location": "Three Geometrical Problems", "latex": "SOP = \\frac{1}{3}SOA", "name": null, "statement": "Pappus's hyperbola construction trisects the angle SOA, giving the angle SOP as one third of it.", "kind": "result", "symbols": [ { "unit": null, "symbol": "SOP", "meaning": "trisected angle" }, { "unit": null, "symbol": "SOA", "meaning": "the given angle" } ], "sympy": "Eq(SOP, SOA/3)", "physics": false, "states": [], "concepts": [ "concept/hyperbola", "concept/plane-angle", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-00fbf51215", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 263", "location": "Three Geometrical Problems", "latex": "y^2 = \\frac{1}{4}x", "name": null, "statement": "Equation of the parabola intersected with a circle in Descartes's trisection.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" } ], "sympy": "Eq(y**2, x/4)", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/parabola" ] }, { "id": "ball-mathematical-recreations-1905/eq-e1f367ca65", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 263", "location": "Three Geometrical Problems", "latex": "x^2 + y^2 - \\frac{13}{4}x + 4ay = 0", "name": null, "statement": "Equation of the circle intersected with a parabola in Descartes's trisection.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "a", "meaning": "sine of the given angle" } ], "sympy": "Eq(x**2 + y**2 - Rational(13, 4)*x + 4*a*y, 0)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/conic-section" ] }, { "id": "ball-mathematical-recreations-1905/eq-8530060ca9", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 263", "location": "Three Geometrical Problems", "latex": "4y^3 = 3y - a", "name": null, "statement": "The ordinates of the intersection points satisfy this cubic; the smaller positive root is the sine of one third of the angle whose sine is a.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate of an intersection point" }, { "unit": null, "symbol": "a", "meaning": "sine of the given angle" } ], "sympy": "Eq(4*y**3, 3*y - a)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/sine", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-d4ca16ff6f", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 264", "location": "Three Geometrical Problems", "latex": "AH = 2 \\dotm HL", "name": null, "statement": "In Clairaut's construction, AH is twice HL.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "AH", "meaning": "third of the chord AB" }, { "unit": null, "symbol": "HL", "meaning": "segment from H to L on the bisector" } ], "sympy": "Eq(AH, 2*HL)", "physics": false, "states": [], "concepts": [ "concept/bisector", "concept/line-segment", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-c34f53f3c6", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 264", "location": "Three Geometrical Problems", "latex": "AP : PM = AH : HL = 2 : 1", "name": null, "statement": "By the focus and directrix property, the ratio AP to PM equals 2 to 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AP", "meaning": "distance from focus A to P" }, { "unit": null, "symbol": "PM", "meaning": "distance from P to the directrix OC" } ], "sympy": "Eq(AP/PM, 2)", "physics": false, "states": [], "concepts": [ "concept/directrix", "concept/focus", "concept/hyperbola" ] }, { "id": "ball-mathematical-recreations-1905/eq-ce01061e16", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 264", "location": "Three Geometrical Problems", "latex": "AP = 2 \\dotm PM = PQ", "name": null, "statement": "AP is twice PM, and PM is half of PQ.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AP", "meaning": "distance from A to P" }, { "unit": null, "symbol": "PM", "meaning": "distance from P to the directrix" }, { "unit": null, "symbol": "PQ", "meaning": "chord from P to Q" } ], "sympy": "Eq(AP, 2*PM)", "physics": false, "states": [], "concepts": [ "concept/chord", "concept/directrix", "concept/focus" ] }, { "id": "ball-mathematical-recreations-1905/eq-31d746161e", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 264", "location": "Three Geometrical Problems", "latex": "AP = PQ = QR", "name": null, "statement": "By symmetry the three chords AP, PQ and QR are equal, so the angles AOP, POQ and QOR are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AP", "meaning": "chord from A to P" }, { "unit": null, "symbol": "PQ", "meaning": "chord from P to Q" }, { "unit": null, "symbol": "QR", "meaning": "chord from Q to R" } ], "sympy": "Eq(AP, PQ)", "physics": false, "states": [], "concepts": [ "concept/chord", "concept/symmetry", "concept/trisection-of-an-angle" ] }, { "id": "ball-mathematical-recreations-1905/eq-f308c3e8d0", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 269", "location": "Three Geometrical Problems", "latex": "6336/2017\\frac14 <\\pi<14688/4673\\frac12", "name": null, "statement": "Archimedes' bounds from the 96-sided polygons: pi lies between 6336/2017 1/4 and 14688/4673 1/2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of the circumference of a circle to its diameter" } ], "sympy": "And(Lt(25344/8069, pi), Lt(pi, 29376/9347))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/inequality", "concept/regular-polygon", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-0526c3a1a6", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 269", "location": "Three Geometrical Problems", "latex": "\\sin\\theta < \\theta < \\tan\\theta", "name": null, "statement": "The proposition Archimedes' polygon method is equivalent to: sine of an angle is less than the angle, which is less than its tangent.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "angle" } ], "sympy": "And(Lt(sin(theta), theta), Lt(theta, tan(theta)))", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/plane-angle", "concept/sine", "concept/tangent-function" ] }, { "id": "ball-mathematical-recreations-1905/eq-1dcafac6c4", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 269", "location": "Three Geometrical Problems", "latex": "\\theta= \\pi/96", "name": null, "statement": "In Archimedes' method the angle θ is set equal to π/96.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "angle used in Archimedes' 96-sided polygon method" }, { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(theta, pi/96)", "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/regular-polygon", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-d5e6f70c02", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 269", "location": "Three Geometrical Problems", "latex": "\\pi = 3^{\\circ} 8' 30''", "name": null, "statement": "Ptolemy asserted that pi equals 3 degrees 8 minutes 30 seconds.", "kind": "result", "symbols": [ { "unit": "degree, minute, second", "symbol": "π", "meaning": "ratio of circumference to diameter, as Ptolemy gives it in sexagesimal angle units" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/plane-angle", "quantity/pi", "unit/unit" ] }, { "id": "ball-mathematical-recreations-1905/eq-86cf3cf683", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 269", "location": "Three Geometrical Problems", "latex": "\\pi = 3 + \\frac8{60} + \\frac{30}{3600} =\\allowbreak 3\\frac{17}{120} =\\allowbreak 3.141\\dot6", "name": null, "statement": "Ptolemy's sexagesimal value written as a decimal fraction, 3 17/120 = 3.1416 approximately.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi, 3 + Rational(8,60) + Rational(30,3600))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/rational-number", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-0b59413ddd", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 270", "location": "Three Geometrical Problems", "latex": "b^2=\\frac{1}{2}-\\frac{1}{2}(1-a^2)^{\\frac{1}{2}}", "name": null, "statement": "Aryabhata's relation: the squared side b of the inscribed 2n-gon in a circle of unit diameter follows from the side a of the inscribed n-gon.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of a regular inscribed polygon of n sides in a circle of unit diameter" }, { "unit": null, "symbol": "b", "meaning": "side of the regular inscribed polygon of 2n sides" }, { "unit": null, "symbol": "n", "meaning": "number of sides of the first polygon" } ], "sympy": "Eq(b**2, Rational(1,2) - Rational(1,2)*(1 - a**2)**Rational(1,2))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/regular-polygon", "concept/side", "concept/square" ] }, { "id": "ball-mathematical-recreations-1905/eq-bc44af1f24", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 272", "location": "Three Geometrical Problems", "latex": "2 \\sin^2\\frac{1}{2}\\theta = 1-\\cos \\theta", "name": null, "statement": "Half-angle identity used by Vieta to halve polygon sides repeatedly.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "angle" } ], "sympy": "Eq(2*sin(theta/2)**2, 1 - cos(theta))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/plane-angle", "concept/sine" ] }, { "id": "ball-mathematical-recreations-1905/eq-f9900663f0", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 272", "location": "Three Geometrical Problems", "latex": "\\frac{2}{\\pi} = \\frac{\\surd 2}{2} \\frac{\\surd (2+\\surd 2)}{2} \\frac{\\surd \\{2+\\surd (2+\\surd 2)\\}}{2} \\dotsm\\;", "name": "Vieta's product", "statement": "Vieta's infinite product giving 2/π as a product of nested square roots.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": null, "physics": false, "states": [ "theorem/vieta-s-product" ], "concepts": [ "concept/approximation", "concept/infinite-sequence", "concept/surd", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-5da534babd", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 272", "location": "Three Geometrical Problems", "latex": "1-\\cos A = 2 \\sin^2\\frac{1}{2}A", "name": null, "statement": "Identity van Ceulen used to double polygon sides when computing perimeters.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle" } ], "sympy": "Eq(1 - cos(A), 2*sin(A/2)**2)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/plane-angle", "concept/sine" ] }, { "id": "ball-mathematical-recreations-1905/eq-c76ccfe2f2", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 273", "location": "Three Geometrical Problems", "latex": "3 \\sin\\theta /(2 + \\cos\\theta) < \\theta < (2 \\sin\\frac{1}{3}\\theta + \\tan\\frac{1}{3}\\theta)", "name": "Snell's theorem", "statement": "Snell's bounds on an angle (and so on an arc), by which a polygon of n sides gives pi to more places than Archimedes' method.", "kind": "result", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "angle subtending the arc" } ], "sympy": "And(Lt(3*sin(theta)/(2 + cos(theta)), theta), Lt(theta, 2*sin(theta/3) + tan(theta/3)))", "physics": false, "states": [ "theorem/snell-s-theorem" ], "concepts": [ "concept/approximation", "concept/cosine", "concept/inequality", "concept/plane-angle", "concept/sine", "concept/tangent-function", "quantity/arc-of-a-circle" ] }, { "id": "ball-mathematical-recreations-1905/eq-21b4497145", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 274", "location": "Three Geometrical Problems", "latex": "\\frac{\\pi}{2}=\\frac{2\\dotm 2\\dotm 4\\dotm 4\\dotm 6\\dotm 6\\dotsm} {1\\dotm 3\\dotm 3\\dotm 5\\dotm 5\\dotm 7\\dotm 7\\dotsm}", "name": "Wallis's product", "statement": "Wallis's infinite product for pi/2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": null, "physics": false, "states": [ "theorem/wallis-s-product" ], "concepts": [ "concept/infinite-sequence", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-90497d531d", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 274", "location": "Three Geometrical Problems", "latex": "\\frac{\\pi}{4}=1+\\frac{1^2}{2} \\genfrac{}{}{0pt}{}{}{+} \\frac{3^2}{2} \\genfrac{}{}{0pt}{}{}{+} \\frac{5^2}{2} \\genfrac{}{}{0pt}{}{}{+}\\ldots\\;", "name": "Brouncker's series", "statement": "Brouncker's continued-fraction-type series for pi/4, as quoted by Wallis.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": null, "physics": false, "states": [ "theorem/brouncker-s-series" ], "concepts": [ "concept/infinite-sequence", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-93416b2378", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 275", "location": "Three Geometrical Problems", "latex": "\\theta = \\tan\\theta - \\frac{1}{3}\\tan^3\\theta + \\frac{1}{5}\\tan^5\\theta - \\dotsb", "name": "Gregory's series", "statement": "Gregory's arctangent series, true only for θ between minus π/4 and π/4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "angle" } ], "sympy": null, "physics": false, "states": [ "theorem/arctangent-series" ], "concepts": [ "concept/infinite-sequence", "concept/plane-angle", "concept/tangent-function" ] }, { "id": "ball-mathematical-recreations-1905/eq-6fb399cf50", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 275", "location": "Three Geometrical Problems", "latex": "\\tfrac{1}{4}\\pi = 4 \\tan^{-1}\\tfrac{1}{5} -\\tan^{-1}\\tfrac{1}{239}", "name": "Machin's formula", "statement": "Machin's arctangent formula for pi/4, used to compute pi to 100 places.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(Rational(1,4)*pi, 4*atan(Rational(1,5)) - atan(Rational(1,239)))", "physics": false, "states": [ "theorem/machin-s-formula-for-pi" ], "concepts": [ "quantity/pi", "theorem/arctangent-series" ] }, { "id": "ball-mathematical-recreations-1905/eq-4e6e9e4e4d", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 275", "location": "Three Geometrical Problems", "latex": "\\frac{1}{4}\\pi =\\allowbreak \\tan^{-1}\\frac{1}{2} +\\allowbreak \\tan^{-1}\\frac{1}{3}", "name": null, "statement": "Hutton's arctangent formula for pi/4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi/4, atan(Rational(1,2)) + atan(Rational(1,3)))", "physics": false, "states": [], "concepts": [ "quantity/pi", "theorem/arctangent-series" ] }, { "id": "ball-mathematical-recreations-1905/eq-55e1596d7e", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 275", "location": "Three Geometrical Problems", "latex": "\\frac{1}{4}\\pi = 5 \\tan^{-1}\\frac{1}{7} + 2\\tan^{-1}\\frac{3}{79}", "name": null, "statement": "Euler's arctangent formula for pi/4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi/4, 5*atan(Rational(1,7)) + 2*atan(Rational(3,79)))", "physics": false, "states": [], "concepts": [ "quantity/pi", "theorem/arctangent-series" ] }, { "id": "ball-mathematical-recreations-1905/eq-808b1f2f78", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 276", "location": "Three Geometrical Problems", "latex": "\\frac{1}{4}\\pi=4\\tan^{-1}\\frac{1}{5} - \\tan^{-1}\\frac{1}{70} + \\tan^{-1}\\frac{1}{99}", "name": null, "statement": "Rutherford's arctangent formula for pi/4, used for his 1841 calculation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi/4, 4*atan(Rational(1,5)) - atan(Rational(1,70)) + atan(Rational(1,99)))", "physics": false, "states": [], "concepts": [ "quantity/pi", "theorem/arctangent-series" ] }, { "id": "ball-mathematical-recreations-1905/eq-533530bbd6", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 276", "location": "Three Geometrical Problems", "latex": "\\frac{1}{4}\\pi= \\tan^{-1}\\frac{1}{2} +\\tan^{-1}\\frac{1}{5} + \\tan^{-1}\\frac{1}{8}", "name": null, "statement": "Dase's arctangent formula for pi/4, used for his 1844 calculation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi/4, atan(Rational(1,2)) + atan(Rational(1,5)) + atan(Rational(1,8)))", "physics": false, "states": [], "concepts": [ "quantity/pi", "theorem/arctangent-series" ] }, { "id": "ball-mathematical-recreations-1905/eq-315a02ca7e", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 276", "location": "Three Geometrical Problems", "latex": "\\frac{1}{4}\\pi = 2\\tan^{-1}\\frac{1}{3} + \\tan^{-1}\\frac{1}{7}", "name": null, "statement": "Clausen's arctangent formula for pi/4, used for his 1847 calculation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi/4, 2*atan(Rational(1,3)) + atan(Rational(1,7)))", "physics": false, "states": [], "concepts": [ "quantity/pi", "theorem/arctangent-series" ] }, { "id": "ball-mathematical-recreations-1905/eq-b4b2eccd75", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 277", "location": "Three Geometrical Problems", "latex": "\\frac{\\pi}{6} = \\frac{1}{2} + \\frac{1}{2}\\dotm \\frac{1}{3\\dotm 2^3} + \\frac{1\\dotm 3}{2\\dotm 4} \\dotm \\frac{1}{5\\dotm 2^5} + \\dotsb\\;", "name": null, "statement": "A rapidly converging series for pi/6 quoted in the chapter.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-2255548018", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 277", "location": "Three Geometrical Problems", "latex": "\\frac{\\pi}{4} = 2 + 22\\tan^{-1}\\frac{1}{28} + \\tan^{-1}\\frac{1}{443} - 5\\tan^{-1}\\frac{1}{1393} - 10\\tan^{-1}\\frac{1}{11018}\\;", "name": null, "statement": "Escott's arctangent formula for pi/4 as printed in the book; a rough hand estimate did not reproduce pi/4 from these coefficients, so it needs a numerical check (possible misprint in the source).", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi/4, 2 + 22*atan(Rational(1,28)) + atan(Rational(1,443)) - 5*atan(Rational(1,1393)) - 10*atan(Rational(1,11018)))", "physics": false, "states": [], "concepts": [ "quantity/pi", "theorem/arctangent-series" ] }, { "id": "ball-mathematical-recreations-1905/eq-8f20de365f", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 275", "location": "Three Geometrical Problems", "latex": "\\tan^{-1}\\tfrac{1}{5}", "name": null, "statement": "Placeholder check, not an equation: omitted.", "kind": "result", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [] }, { "id": "ball-mathematical-recreations-1905/eq-b4b20813d9", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 271", "location": "Three Geometrical Problems", "latex": "\\frac{3}{4}(\\sqrt{3} + \\sqrt{6})", "name": null, "statement": "Cusa's believed value of pi, which the chapter reports is about 3.1423 and which Regiomontanus is said to have shown wrong.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi, Rational(3,4)*(sqrt(3) + sqrt(6)))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/surd", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-bbb167059b", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 277", "location": "Three Geometrical Problems", "latex": "2l/\\pi a", "name": null, "statement": "Probability that a stick of length l dropped on a plane ruled with lines a apart lies across a line; used to estimate pi experimentally.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "l", "meaning": "length of the stick" }, { "unit": null, "symbol": "a", "meaning": "distance apart of the ruled parallel lines" }, { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/parallel-lines", "concept/probability", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-d6fa02cc07", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 278", "location": "Three Geometrical Problems", "latex": "6/\\pi^2 = 154/250", "name": null, "statement": "Relating the probability 6/π² that two random numbers are prime to each other to the observed frequency 154/250 gives π about 3.12.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(6/pi**2, Rational(154,250))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/probability", "concept/rational-number", "concept/relatively-prime", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-3fcabcbcc4", "chapter": "ball-mathematical-recreations-1905/ch-viii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 278", "location": "Three Geometrical Problems", "latex": "\\pi=3.12", "name": null, "statement": "The value of pi that the chapter derives from the coprime-pair experiment.", "kind": "result", "symbols": [ { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi, 3.12)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/probability", "quantity/pi" ] }, { "id": "ball-mathematical-recreations-1905/eq-2f7f1c04ff", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 282", "location": "Mersenne's Numbers", "latex": "N=2^p - 1", "name": "Mersenne number", "statement": "N, the number used in this chapter, is defined as 2 to the power p minus 1, for a prime p.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "N", "meaning": "a number of the form 2^p-1 (a Mersenne number)" }, { "unit": null, "symbol": "p", "meaning": "the exponent, taken to be prime" } ], "sympy": "Eq(N, 2**p - 1)", "physics": false, "states": [ "concept/mersenne-number" ], "concepts": [ "concept/power", "concept/prime-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-55158e48f1", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 283", "location": "Mersenne's Numbers", "latex": "2^{4n+3}-1 \\equiv 0 \\pmod{8n + 7}", "name": "Euler's theorem on factors of Mersenne numbers", "statement": "If 4n+3 and 8n+7 are both prime, then 2^(4n+3) - 1 is divisible by 8n+7, so 8n+7 is a factor of the Mersenne number for p = 4n+3 (Euler's proposition, proved by Lagrange in 1775).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "an integer" } ], "sympy": "Eq(Mod(2**(4*n+3) - 1, 8*n + 7), 0)", "physics": false, "states": [ "theorem/euler-s-theorem-on-factors-of-mersenne-numbers" ], "concepts": [ "concept/congruence", "concept/divisibility", "concept/mersenne-number", "concept/prime-factor", "quantity/modulus-of-a-complex-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-a5c51aa7c0", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 289", "location": "Mersenne's Numbers", "latex": "N = n^2 + a = (n + b)^2 - (b^2 + 2bn - a)", "name": "difference of two squares representation of N", "statement": "Since N can be written as n squared plus a, it can also be written as the difference of the squares (n+b)^2 and (b^2 + 2bn - a), which is the starting point for the indeterminate-equation method.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "N", "meaning": "a number of the form 2^p-1" }, { "unit": null, "symbol": "n", "meaning": "an integer chosen from N" }, { "unit": null, "symbol": "a", "meaning": "the remainder N - n^2" }, { "unit": null, "symbol": "b", "meaning": "a number added to n in the square (n + b)^2" } ], "sympy": "Eq(N, n**2 + a)", "physics": false, "states": [ "theorem/difference-of-two-squares-representation-of-n" ], "concepts": [ "concept/factor", "concept/indeterminate-equation", "concept/mersenne-number", "theorem/difference-of-two-squares" ] }, { "id": "ball-mathematical-recreations-1905/eq-4bb1fa499b", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 289", "location": "Mersenne's Numbers", "latex": "x^2 = (2py + H)^2 - 4(K - y)", "name": null, "statement": "An indeterminate equation in integers x and y whose integral solutions with y < K give values of u and v, and hence factors of N.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "an integer variable of the indeterminate equation" }, { "unit": null, "symbol": "y", "meaning": "an integer variable of the indeterminate equation, required to be less than K" }, { "unit": null, "symbol": "p", "meaning": "the prime exponent of the Mersenne number" }, { "unit": null, "symbol": "H", "meaning": "a number that can be calculated from N" }, { "unit": null, "symbol": "K", "meaning": "a number that can be calculated from N" } ], "sympy": "Eq(x**2, (2*p*y + H)**2 - 4*(K - y))", "physics": false, "states": [], "concepts": [ "concept/indeterminate-equation", "concept/integral", "concept/mersenne-number", "theorem/difference-of-two-squares" ] }, { "id": "ball-mathematical-recreations-1905/eq-72c0863598", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 290", "location": "Mersenne's Numbers", "latex": "4s + t = \\alpha + 8px", "name": null, "statement": "The first of two linked conditions obtained from writing (2pt+1)(8ps+1) = 2^p - 1, giving a second indeterminate equation for the factor parameters s and t.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s", "meaning": "an integer parameter of the factor 8ps+1" }, { "unit": null, "symbol": "t", "meaning": "an odd integer parameter of the factor 2pt+1" }, { "unit": null, "symbol": "alpha", "meaning": "the quantity (2^(p-1) - 1)/p modulo the term 8p beta" }, { "unit": null, "symbol": "beta", "meaning": "a quantity defined with alpha" }, { "unit": null, "symbol": "x", "meaning": "an integer, not greater than beta" }, { "unit": null, "symbol": "p", "meaning": "the prime exponent" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/factor", "concept/indeterminate-equation", "concept/integer", "concept/mersenne-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-66c3f34b16", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 290", "location": "Mersenne's Numbers", "latex": "st = \\beta - x", "name": null, "statement": "The second of the two linked conditions for the factor parameters s and t, with x not greater than beta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s", "meaning": "an integer parameter of the factor 8ps+1" }, { "unit": null, "symbol": "t", "meaning": "an odd integer parameter of the factor 2pt+1" }, { "unit": null, "symbol": "beta", "meaning": "a quantity defined with alpha" }, { "unit": null, "symbol": "x", "meaning": "an integer, not greater than beta" } ], "sympy": "Eq(s*t, beta - x)", "physics": false, "states": [], "concepts": [ "concept/factor", "concept/indeterminate-equation" ] }, { "id": "ball-mathematical-recreations-1905/eq-ff44fa9e6e", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 292", "location": "Mersenne's Numbers", "latex": "2^{2pt}-1 \\equiv 0", "name": "Fermat's Theorem", "statement": "By Fermat's Theorem, when 2pt+1 is prime, 2 raised to the power 2pt minus 1 is divisible by 2pt+1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "the prime exponent of the Mersenne number" }, { "unit": null, "symbol": "t", "meaning": "a positive integer such that 2pt+1 is prime" } ], "sympy": "Eq(Mod(2**(2*p*t) - 1, 2*p*t + 1), 0)", "physics": false, "states": [ "theorem/fermat-s-little-theorem" ], "concepts": [ "concept/congruence", "concept/divisibility", "quantity/modulus-of-a-complex-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-c38cca4223", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 292", "location": "Mersenne's Numbers", "latex": "(2^p-1)(1+2^p+2^{2p}+\\dotsb+2^{(2t-1)p}) \\equiv 0", "name": null, "statement": "Modulo the prime 2pt+1, the product of 2^p-1 and the geometric sum of powers of 2^p is zero, so a factor of 2^p-1 is found when the second factor is prime to 2pt+1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "the prime exponent of the Mersenne number" }, { "unit": null, "symbol": "t", "meaning": "a positive integer such that 2pt+1 is prime" } ], "sympy": "Eq(Mod((2**p - 1)*(1 + sum(2**(j*p) for j in range(0, 2*t))), 2*p*t + 1), 0)", "physics": false, "states": [], "concepts": [ "concept/congruence", "concept/divisibility", "concept/factor", "concept/mersenne-number", "theorem/fermat-s-little-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-181d323397", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 292", "location": "Mersenne's Numbers", "latex": "(2^p-1)(2^p + 1)\\equiv 0", "name": "Euler's theorem of 1732", "statement": "With t = 1 and 2p+1 prime, the product (2^p - 1)(2^p + 1) vanishes modulo 2p+1, the basis of Euler's 1732 theorem.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "a prime exponent such that 2p+1 is prime" } ], "sympy": "Eq(Mod((2**p - 1)*(2**p + 1), 2*p + 1), 0)", "physics": false, "states": [ "theorem/euler-s-theorem-on-factors-of-mersenne-numbers" ], "concepts": [ "concept/congruence", "concept/mersenne-number", "theorem/difference-of-two-squares" ] }, { "id": "ball-mathematical-recreations-1905/eq-5bd61aa02c", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 292", "location": "Mersenne's Numbers", "latex": "2^p\\equiv1", "name": null, "statement": "Modulo 2p+1, 2^p is congruent to 1 when 2^p + 1 is prime to 2p+1, which holds for p = 4m + 3, so 2p+1 divides N.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "a prime exponent of the form 4m+3" } ], "sympy": "Eq(Mod(2**p, 2*p + 1), 1)", "physics": false, "states": [], "concepts": [ "concept/congruence", "concept/prime-factor", "concept/relatively-prime", "theorem/euler-s-theorem-on-factors-of-mersenne-numbers" ] }, { "id": "ball-mathematical-recreations-1905/eq-33ce914910", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 290", "location": "Mersenne's Numbers", "latex": "2^{p+y} \\equiv z", "name": null, "statement": "The Canon Arithmeticus method: a prime q and exponents y and z are sought with 2 to the power p+y congruent to z modulo q.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "q", "meaning": "an odd prime used as modulus" }, { "unit": null, "symbol": "y", "meaning": "an exponent chosen at will" }, { "unit": null, "symbol": "z", "meaning": "a residue chosen at will" }, { "unit": null, "symbol": "p", "meaning": "the prime exponent of the Mersenne number" } ], "sympy": "Eq(Mod(2**(p + y), q), z)", "physics": false, "states": [], "concepts": [ "concept/congruence", "concept/power", "concept/table-of-power-residues" ] }, { "id": "ball-mathematical-recreations-1905/eq-ab57689dd0", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 290", "location": "Mersenne's Numbers", "latex": "2^y (2^p - 1) \\equiv 0", "name": null, "statement": "Combining the two congruences modulo q gives 2^y times (2^p - 1) congruent to zero, from which the book concludes that q divides N.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "an exponent chosen at will" }, { "unit": null, "symbol": "p", "meaning": "the prime exponent of the Mersenne number" } ], "sympy": "Eq(Mod(2**y*(2**p - 1), q), 0)", "physics": false, "states": [], "concepts": [ "concept/congruence", "concept/divisibility", "concept/factor" ] }, { "id": "ball-mathematical-recreations-1905/eq-48b26d669b", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 286", "location": "Mersenne's Numbers", "latex": "2^p = 1", "name": null, "statement": "The book concludes that 2^p = 1 from the preceding congruences, so q is a divisor of N. The printed sign appears to be a transcription loss: the argument requires the congruence 2^p ≡ 1 (mod q), not an equality, which is how it is read here.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "the prime exponent of the Mersenne number" } ], "sympy": "Eq(2**p, 1)", "physics": false, "states": [], "concepts": [ "concept/congruence", "concept/divisibility", "concept/factor" ] }, { "id": "ball-mathematical-recreations-1905/eq-05fce4282e", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 291", "location": "Mersenne's Numbers", "latex": "2^{u-v} \\equiv 1", "name": null, "statement": "If the remainders of powers of 2 agree for exponents u and v modulo a prime q, then 2 to the power u minus v is congruent to 1 modulo q.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "an exponent for which 2^u has a stated remainder modulo q" }, { "unit": null, "symbol": "v", "meaning": "an exponent for which 2^v has the same remainder modulo q" }, { "unit": null, "symbol": "q", "meaning": "a prime modulus of the form 2pt+1" } ], "sympy": "Eq(Mod(2**(u - v), q), 1)", "physics": false, "states": [], "concepts": [ "concept/congruence", "concept/power", "concept/remainder" ] }, { "id": "ball-mathematical-recreations-1905/eq-c5755f26a3", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 291", "location": "Mersenne's Numbers", "latex": "2^u \\equiv 2^v", "name": null, "statement": "If the remainders of 2^x modulo a prime q agree for x = u and x = v, then 2^u is congruent to 2^v modulo q.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "an exponent" }, { "unit": null, "symbol": "v", "meaning": "an exponent" } ], "sympy": "Eq(Mod(2**u, q), Mod(2**v, q))", "physics": false, "states": [], "concepts": [ "concept/congruence", "concept/power", "concept/remainder" ] }, { "id": "ball-mathematical-recreations-1905/eq-75e76307c1", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 282", "location": "Mersenne's Numbers", "latex": "2047 = 23 \\times 89", "name": null, "statement": "The Mersenne number for p = 11 factors as 23 times 89, so it is composite.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "the exponent 11" } ], "sympy": "Eq(2047, 23*89)", "physics": false, "states": [], "concepts": [ "concept/factor", "concept/mersenne-number", "concept/prime-factor", "concept/prime-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-5aee7c8c4b", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 282", "location": "Mersenne's Numbers", "latex": "137438953471 = 223 \\times 616318177", "name": null, "statement": "The Mersenne number for p = 37 factors as 223 times 616318177, so it is composite; the factorization was given by Fermat.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "the exponent 37" } ], "sympy": "Eq(137438953471, 223*616318177)", "physics": false, "states": [], "concepts": [ "concept/factor", "concept/mersenne-number", "concept/prime-factor", "theorem/fermat-s-little-theorem" ] }, { "id": "ball-mathematical-recreations-1905/eq-c030590cc2", "chapter": "ball-mathematical-recreations-1905/ch-ix", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 286", "location": "Mersenne's Numbers", "latex": "8i \\pm 1", "name": null, "statement": "Any prime factor of a Mersenne number must be of one of the forms 8i plus or minus 1, since N is of the form 2A^2 minus B^2 with A even and B odd.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "i", "meaning": "an integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/congruence", "concept/integer", "concept/prime-factor", "concept/quadratic-residue" ] }, { "id": "ball-mathematical-recreations-1905/eq-0e9e3b5a0b", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 337", "location": "Cryptographs and Ciphers", "latex": "10^5", "name": null, "statement": "The five-digit code dictionary has 10^5 numbers available, one for each possible five-digit code word.", "kind": "result", "symbols": [], "sympy": "Eq(N, 10**5)", "physics": false, "states": [], "concepts": [ "concept/book-cipher", "concept/digit", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-926aad6674", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 338", "location": "Cryptographs and Ciphers", "latex": "26^4", "name": null, "statement": "Four letters of the twenty-six-letter alphabet give 26^4 (456976) possible variations for each word or phrase.", "kind": "result", "symbols": [], "sympy": "Eq(V, 26**4)", "physics": false, "states": [], "concepts": [ "concept/alphabet", "concept/cipher", "concept/power", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-f7daf7d70b", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 338", "location": "Cryptographs and Ciphers", "latex": "36^3", "name": null, "statement": "Three symbols each chosen from the ten digits and the twenty-six letters give 36^3 (46656) possible words.", "kind": "result", "symbols": [], "sympy": "Eq(V, 36**3)", "physics": false, "states": [], "concepts": [ "concept/alphabet", "concept/digit", "concept/power", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-6d3380a388", "chapter": "ball-mathematical-recreations-1905/ch-xi", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 337", "location": "Cryptographs and Ciphers", "latex": "5n", "name": null, "statement": "A message of n words is written as 5n digits, since each word is represented by a five-digit number.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of words in the message" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/book-cipher", "concept/digit", "concept/real-number" ] }, { "id": "ball-mathematical-recreations-1905/eq-ed7d62a9b1", "chapter": "ball-mathematical-recreations-1905/ch-xii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 348", "location": "Hyper-space", "latex": "\\phi(x, y, z) = 0", "name": null, "statement": "The boundary of a solid in ordinary three-dimensional space is the set of points whose co-ordinates satisfy phi(x, y, z) = 0.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function whose zero set gives the boundary of a solid" }, { "unit": null, "symbol": "x", "meaning": "co-ordinate of a point in space" }, { "unit": null, "symbol": "y", "meaning": "co-ordinate of a point in space" }, { "unit": null, "symbol": "z", "meaning": "co-ordinate of a point in space" } ], "sympy": "Eq(phi(x, y, z), 0)", "physics": false, "states": [], "concepts": [ "concept/coordinate-geometry", "concept/surface", "concept/three-dimensional-figure" ] }, { "id": "ball-mathematical-recreations-1905/eq-cae42653b7", "chapter": "ball-mathematical-recreations-1905/ch-xii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 348", "location": "Hyper-space", "latex": "\\phi(x, y, z, \\omega) = 0", "name": null, "statement": "In four dimensions the boundary of a body is the set of points whose co-ordinates satisfy phi(x, y, z, omega) = 0.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function whose zero set gives the boundary of a body in four dimensions" }, { "unit": null, "symbol": "x", "meaning": "co-ordinate of a point in four-dimensional space" }, { "unit": null, "symbol": "y", "meaning": "co-ordinate of a point in four-dimensional space" }, { "unit": null, "symbol": "z", "meaning": "co-ordinate of a point in four-dimensional space" }, { "unit": null, "symbol": "omega", "meaning": "co-ordinate in the fourth dimension" } ], "sympy": "Eq(phi(x, y, z, omega), 0)", "physics": false, "states": [], "concepts": [ "concept/coordinate-geometry", "concept/dimension", "concept/hyper-space", "concept/surface" ] }, { "id": "ball-mathematical-recreations-1905/eq-5522dd5ee8", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 363", "location": "Time and its Measurement", "latex": "a = N - 4 \\{N/4\\}", "name": null, "statement": "The remainder a, when the year N is divided by 4, equals N minus 4 times the integral part of N/4.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "remainder of the year number on division by 4" }, { "unit": null, "symbol": "N", "meaning": "the number of the year" }, { "unit": null, "symbol": "\\{N/x\\}", "meaning": "integral part of the quotient when N is divided by x" } ], "sympy": "Eq(a, N - 4*floor(N/4))", "physics": false, "states": [], "concepts": [ "concept/integer-part", "concept/remainder", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-3f7c290c16", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 363", "location": "Time and its Measurement", "latex": "b = N - 7 \\{N/7\\}", "name": null, "statement": "The remainder b, when the year N is divided by 7, equals N minus 7 times the integral part of N/7.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "b", "meaning": "remainder of the year number on division by 7" }, { "unit": null, "symbol": "N", "meaning": "the number of the year" } ], "sympy": "Eq(b, N - 7*floor(N/7))", "physics": false, "states": [], "concepts": [ "concept/integer-part", "concept/remainder", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-8d71b26aac", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 363", "location": "Time and its Measurement", "latex": "c = N - 19 \\{N/19\\}", "name": null, "statement": "The remainder c, when the year N is divided by 19, equals N minus 19 times the integral part of N/19.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c", "meaning": "remainder of the year number on division by 19" }, { "unit": null, "symbol": "N", "meaning": "the number of the year" } ], "sympy": "Eq(c, N - 19*floor(N/19))", "physics": false, "states": [], "concepts": [ "concept/golden-number", "concept/integer-part", "concept/remainder", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-60aefbe501", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 363", "location": "Time and its Measurement", "latex": "\\xi = \\{N/100\\}\\allowbreak - \\{N/400\\} - \\{N/300\\}", "name": null, "statement": "In the Gregorian calendar the correction xi for the year N is the integral part of N/100 minus the integral part of N/400 minus the integral part of N/300.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "correction term for the Easter constant m in the Gregorian calendar" }, { "unit": null, "symbol": "N", "meaning": "the number of the year" } ], "sympy": "Eq(xi, floor(N/100) - floor(N/400) - floor(N/300))", "physics": false, "states": [], "concepts": [ "concept/gregorian-calendar", "concept/integer-part", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-9ae461c7d0", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 363", "location": "Time and its Measurement", "latex": "\\eta = \\{N/100\\}\\allowbreak - \\{N/400\\} - 2", "name": null, "statement": "In the Gregorian calendar the correction eta for the year N is the integral part of N/100 minus the integral part of N/400 minus 2.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "eta", "meaning": "correction term for the Easter constant n in the Gregorian calendar" }, { "unit": null, "symbol": "N", "meaning": "the number of the year" } ], "sympy": "Eq(eta, floor(N/100) - floor(N/400) - 2)", "physics": false, "states": [], "concepts": [ "concept/gregorian-calendar", "concept/integer-part", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-13cd4ecc8b", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 363", "location": "Time and its Measurement", "latex": "\\xi = 0", "name": null, "statement": "In the Julian calendar the correction xi is zero.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "correction term for the Easter constant m" } ], "sympy": "Eq(xi, 0)", "physics": false, "states": [], "concepts": [ "concept/julian-calendar", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-983eed9437", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 363", "location": "Time and its Measurement", "latex": "\\eta = 0", "name": null, "statement": "In the Julian calendar the correction eta is zero.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "eta", "meaning": "correction term for the Easter constant n" } ], "sympy": "Eq(eta, 0)", "physics": false, "states": [], "concepts": [ "concept/julian-calendar", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-a66d643746", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 362", "location": "Time and its Measurement", "latex": "m=15", "name": null, "statement": "In the Julian calendar the Easter constant m equals 15.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "m", "meaning": "Easter constant for the calendar period (remainder when 15 + xi is divided by 30)" } ], "sympy": "Eq(m, 15)", "physics": false, "states": [], "concepts": [ "concept/julian-calendar", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-167c7be436", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 362", "location": "Time and its Measurement", "latex": "n = 6", "name": null, "statement": "In the Julian calendar the Easter constant n equals 6.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "n", "meaning": "Easter constant for the calendar period (remainder when 6 + eta is divided by 7)" } ], "sympy": "Eq(n, 6)", "physics": false, "states": [], "concepts": [ "concept/julian-calendar", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-42a7585dea", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 362", "location": "Time and its Measurement", "latex": "m = 22", "name": null, "statement": "For the years 1582 to 1699 inclusive in the Gregorian calendar, the Easter constant m equals 22.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "m", "meaning": "Easter constant for the calendar period" } ], "sympy": "Eq(m, 22)", "physics": false, "states": [], "concepts": [ "concept/gregorian-calendar", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-0d33326592", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 362", "location": "Time and its Measurement", "latex": "n = 2", "name": null, "statement": "For the years 1582 to 1699 inclusive in the Gregorian calendar, the Easter constant n equals 2.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "n", "meaning": "Easter constant for the calendar period" } ], "sympy": "Eq(n, 2)", "physics": false, "states": [], "concepts": [ "concept/gregorian-calendar", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-a18095a9e4", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 362", "location": "Time and its Measurement", "latex": "m = 23", "name": null, "statement": "For the years 1700 to 1799 in the Gregorian calendar, the Easter constant m equals 23.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "m", "meaning": "Easter constant for the calendar period" } ], "sympy": "Eq(m, 23)", "physics": false, "states": [], "concepts": [ "concept/gregorian-calendar", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-773083705f", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 362", "location": "Time and its Measurement", "latex": "n = 3", "name": null, "statement": "For the years 1700 to 1799 in the Gregorian calendar, the Easter constant n equals 3.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "n", "meaning": "Easter constant for the calendar period" } ], "sympy": "Eq(n, 3)", "physics": false, "states": [], "concepts": [ "concept/gregorian-calendar", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-60b73e758c", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 362", "location": "Time and its Measurement", "latex": "c > 10", "name": null, "statement": "The exceptional Easter case that moves Easter-day one week earlier applies only when c is greater than 10.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "c", "meaning": "remainder of the year number on division by 19" } ], "sympy": "c > 10", "physics": false, "states": [], "concepts": [ "concept/inequality", "method/date-of-easter" ] }, { "id": "ball-mathematical-recreations-1905/eq-bae39274f2", "chapter": "ball-mathematical-recreations-1905/ch-xiii", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 366", "location": "Time and its Measurement", "latex": "\\sin^{-1} (\\cos \\omega \\sec \\tfrac{1}{2} \\omega) - \\cot^{-1} \\{ \\sin \\omega \\cos(l - \\tfrac{1}{2} \\omega) (\\cos^2 l - \\sin^2 \\omega)^{-\\frac{1}{2}} \\}", "name": null, "statement": "The angle through which the shadow of a normal style moves backwards between sunrise and noon on the longest day, for a dial placed to cut the meridian midway between the equator and the tropic.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "l", "meaning": "latitude of the place" }, { "unit": "degree", "symbol": "omega", "meaning": "obliquity of the ecliptic" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/latitude", "concept/obliquity-of-the-ecliptic", "concept/plane-angle", "instrument/sun-dial" ] }, { "id": "ball-mathematical-recreations-1905/eq-f35fabd7ef", "chapter": "ball-mathematical-recreations-1905/ch-xiv", "book": "ball-mathematical-recreations-1905", "edition": "Macmillan and Co., 4th ed., 1905", "page": "scan 388", "location": "Matter and Ether Theories", "latex": "=10^{12}", "name": null, "statement": "The book fixes its meaning of 'billion' as ten to the twelfth power, following the English usage rather than the French.", "kind": "definition", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/real-number" ] } ], "exercise_sets": [], "problems": [], "errata_statuses": [ "transcriber_marked", "candidate", "probable", "confirmed", "dismissed", "note" ], "errata": [] }