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2X2 squares 3X3 square 4X4 square 5X5 square 6X6 square 8X8 magic square 8X8 square arithmetical order arithmetical sequence arrangement basic row bers breakmove cells in upward center cell central cell Class combination contains corner cells corner diagonal columns cylinder David Eugene Smith diagonal sequence diagrams eight numbers end terms equal equation example follows four Franklin squares Frierson given in Fig hand diagonal harmony horizontal columns illustrate increments initial number Jaina knight's move last numbers left hand magic cubes mathematics method middle cell numbers in Fig odd magic squares pairs of numbers PAUL CARUS perpendicular placed cells Plato Plutarch produce Pythagoras Pythagorean regular rhombus right-hand second cell Section series of numbers shown in Fig shows SIMPLE ALTERNATION square in Fig sub-squares symbols symmetry Tetractys three-figure values Timaeus Totals tt tt tt upper vertical X 4 square
Page 167 - A Hair perhaps divides the False and True ; Yes; and a single Alif were the clue — Could you but find it — to the Treasure-house, And peradventure to THE MASTER too...
Page 202 - Cole, Editor American Mathematical Bulletin, New York. " The selection is well made, the reproduction is handsomely executed, and the brief account which accompanies each portrait is of interest. Prof. Smith has rendered a valuable service to all who have interest in mathematics, by editing this collection. Wherever mathematics is taught, these portraits should adorn the walls ."— William F.
Page 90 - I then confessed to him that in my younger days, having once some leisure (which I still think I might have employed more usefully), I had amused myself in making this kind of magic squares, and at length had acquired such a knack at it that I could fill the cells of any magic square of reasonable size with a series of numbers as fast as I could write them, disposed in such a manner as that the sums of every row, horizontal, perpendicular, or diagonal, should...
Page 100 - ... Also the bent row from 52 to 54, descending to the right, and from 10 to 16, descending to the left, and every one of its parallel bent rows of eight numbers, make 260.
Page 100 - ... hole being cut in a piece of paper of such a size as to take in and show through it just 16 of the little squares, when laid on the greater square, the sum of the 16 numbers so appearing through the hole, wherever it was placed on the greater square, should likewise make 2056.
Page 99 - Also the bent row from 52 descending to 54, and from 43 ascending to 45, and every one of its parallel bent rows of eight numbers, make 260. Also the bent row from 45 to 43, descending to the left, and from 23 to 17, descending to the right, and every one of its parallel bent rows of eight numbers, make 260.
Page 124 - ... .impar numerus immortalis, quia dividi integer non potest, par numerus mortalis, quia dividi potest; licet Varro dicat Pythagoreos putare imparem numerum habere finem, parem esse...
Page 90 - That every straight row, horizontal or vertical, of 8 numbers added together makes 260, and half each row half 260. (2) That the bent row of 8 numbers, ascending and descending diagonally, viz., from 16 ascending to 10, and from 23 descending to 17, and every one of its parallel bent rows of 8 numbers, make 260. Also the bent row from 52...
Page v - ... 197. 1602. There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations. — COMTE, A Positive Philosophy [Martineau], Bk. 1, chap. 1. 1603. Pythagoras says that number is the origin of all things, and certainly the law of number is the key that unlocks the secrets of the universe.
Page vi - Some Final Thoughts There is no science that teaches the harmonies of nature more clearly than mathematics, and the magic squares are like a minor which reflects the symmetry of the divine norm immanent in all things, in the immeasurable immensity of the cosmos and in the construction of the atom not less than in the mysterious depths of the human mind.