The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . ^-ing is one of the most interesting. Let A BCD, AEFG be anytwo squares, placed so thattheir bases may join and formone straight line. Take GHand EK each equal to AB, andjoin HC, CK, KF, FH. Then it may be shewn thatthe triangle HBC is equal inaU respects to the triangle FEK,and the triangle KDC to thetriangle FGH. Therefore thetwo squares are together equiva-lent to the figure CKFH. Itmay then be shewn, with the aid of I. 32, th
The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . ^-ing is one of the most interesting. Let A BCD, AEFG be anytwo squares, placed so thattheir bases may join and formone straight line. Take GHand EK each equal to AB, andjoin HC, CK, KF, FH. Then it may be shewn thatthe triangle HBC is equal inaU respects to the triangle FEK,and the triangle KDC to thetriangle FGH. Therefore thetwo squares are together equiva-lent to the figure CKFH. Itmay then be shewn, with the aid of I. 32, that the figure CKFHis a square. And the side CHia the hypotenuse of a right-angledtriangle of which the sides CB, BH are equal to the sides of thetwo given squares. This demonstration requires no propositionof Euclid after I. 32, and it shews how two given squares maybe cut into pieces which will fit together so as to form a third•quare. Quarterly Journal of Mathematics, Vol. I. A large nimaber of demonstrations of this proposition are col-lected in a dissertation by Job. Jos. Ign. Hoffmann, entitled DmtPythagorischc Lehrsat2,...,Axisgabe. Maim. £UCLIUS ELEMENTS, 267 THE SECOND BOOK. The second book is devoted lo the investigation of relationabetween the rectangles contained by straight lines divided intosegments in various ways. When a stf»!gi=*t line is divided into two parts, each part iscalled a segment by Euclid. It is found convenient to extend themeaning of ^t word segment, and to lay down the following defi-nition. Wv^en a point is taken in a straight line, or in thestraight li^ produced, the distances of the point from the ends ofthe straig ; line are called segments of the straight line. Whe»it is necef!°ary to distinguish them, such segments are called in-temal or eternal, according as the point is in the straight Hne,or in the straight line produced. The student cannot fail to notice that there is an analogybetween the first ten propositions of t
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Keywords: ., bookcentury1800, booksubjectgeometry, booksubjectmathematicsgree
