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A New Hiftorical Relation Tome II.
Of the Indian Method of the Even Squares,
I Thought to have divined it from the examples of the fquares of 16, 36, and 64 cafes, which Agrifpa has given us-
1. As theranksare in even number in the even fquares, they may be confi-dered two by two. Comparing then the firft to the laft, the fecond to the laft fave one, the third to the laft but two, and fo fucceffively, by equally removing us from the firft and the laft ranks, we will call them oppofite, be they tranfverfe, or upright.
Now becaufe that the numbers of one rank, are arithmetically proportional with thofe of another rank of the fame way, it is clear to thofe who underftand arithmetical proportion, that two oppofite ranks do make the fame total fum as two other oppofite ranks, and that if this fum be divided into two equals, each half will be the fum that a Magical rank ought to make.
2. The oppofite numbers are alfo the firft and laft of the whole fquare, the fecond and laft fave one, the third and laft but two, and fo fucceffively, by removing as equally from the firft and laft numbers.; fo that the fom of two oppofite numbers is always equal to the fum of other oppofites.
From hence it is evident, that the numbers oppofite to thofe of one rank, are the numbers which are in the oppofite rank, ana that to render the fums of two oppofite ranks equal, it is neceflary only to take the moity of the numbers of one of the ranks, and to exchange them for their oppofites, which are in the other. As for Example
I 14 If 4
*3 2 3 16
1, 2, 3, 4, do make the firft natural rank of the fquare of 16 cafes, and 13,14a x f, 16, do make the laft rank thereof To render them equal, it is neceflary only to take x and 3, which are the moity ofthe numbers of the firft, and to exchange them for 14 and iy, their oppofites5 and fo 1,14,1;, 4, will make the fum as 13, a, 3, iff.
The tranfverfes between them, and the uprights between them, may reoder themfelves equal by this Method: but becaufe that the choice of the oppofite numbers may be made after feveral ways, the Indians have chofen one,tnatis eafie to retain, which leaves the diameters fuch as they are in the natural (quare, becaufe that they are fuch as they ought to be, and ranges the uprights, when it is intended only to range the tranfverTes. The whole Method confifts then in knowing how to range two oppofite tranfverfes. and the rules are thefe.- - i-
1. They take the half of tne numbers of the upper tranfverfe, and remove them to tne lower: and they take their oppofite numbers in the lower tranfverfe, and remove them to the upper, j :
2. The numbers which remaibJa each rank* do remain there in their natural place, and in their natural order: the tranfprofed do place themfelves every one in the cafe of its oppofite, and confequently la a fubverted order.
3. The firft and the laft numbers of every rank do continue in their natural rank, the fecond and third are tranfprofed, the fourth and the fifth remain, the fixth, and the feventhj are tranfprofed, and fo alternatively two are tranfprofed, and two remain.