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A New Hiftorical Relation Tome II.
that it fuffices that the numbers of a Band be Arithmetically proportional tg thofe of every other Band, I will denote the firft numbers of every Band by the letters of the Latin Alphabet, and the differences between the numbers of the fame Band by the letters of the Greek Alphabet: and to the end that the numbers of a Band be Arithmetically proportional to the numbers of every other Band, I will fet down
the differences of the numbers of each band by the fame Greek Letters.
i. Nothing hinders why the Signa , may not be placed inftead of the Sign -h,either before all the differences, or before fome, provided that the fame Sign be before the fame difference in each band: for fo the Arithmetical proportion will not be altered.
a. The greater a fquare (hall be, the more Latin and Greek. Letters it will have; but every band will never have but one Latin Letter, and all the Gree\ Letters; and the Latin Letter (hall be different in each band. Every bar on the contrary (hall have all the Latin Letters* and all except the firft (hall have a 7rA ^ Letter, which (hall be different in every bar.
FRom hence it follows, i. That the diameters of the augmented fquare have each all the Lam and all the Greeks Letters, becaufe that they have each a cafe of every band, and a cafe of every bar, and that the cafes of every band do-give them all the Latin Letteis, and the cafes of each bar all the Greek. The fum then of thefe two diameters is the fame, viz- that of all the Letters, as well Grtel^ as Latin, taken at once. Now thefe two diameters do make an upright and a tranfverfe in the Magical fquare, becaufe that in the operation of Backet, their fum changes not by the lofs or acquifition of any number, as I have already remarked.
2- As the ranks of the true fquare, whether tranfverfe or upright, are as diftant from their complements, as there are cafes in the fide of the true fquare, it follows that the bands, and the bars, which begin with a complement, or above this complement, touch not, that is to fay, have no cafe at the defective rank of this complement; and that the bands and the bars which begin with a defective rank or above, have no cafe in its complement: the Letters then of the defective rank, are all different from thofe of the complements; becaufe that different bands have different Latin Letters, and that different bars have different Greek,Letters- But becaufe that all the bands, and all the bars, have each % cafe in all the defective ranks, or in their complements a a then every defective rank whatever, will have all the Letters, when it (hall have received its complement ; it will have all the Latin, becaufe that all the bands, pafiing through every defective rank, or through its complement, do there leave all the Latin