About the Book
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1837 Excerpt: ...Slainsfield School. Draw the figure as described in the question; then from each of the right angles ABF, MBC take away the common angle MBA and there remains MBF= ABC hence the triangles MBF, CBA are equal, for they are equiangular, and BF=BA, and in like manner CNK=ABC, consequently MN is parallel to BC and also equal to it, but the lines PN and PM are respectively parallel to CA and BA for they are simply the prolongation of the sides of the squares described on them. Consequently the triangles BAC and MPN ST By the question, the inscribed Di c squares, are in the ratio of 2, 6, 12 and 20, divide the side of the square AB in this ratio in the points L, N, O, and on it describe a semicircle; drop the ordinates LQ, NR and OS, and join BQ, BR, and BS, make BM, BP and BT respectively equal to BQ, BR, BS, and on BM, BP and BT describe squares, then will the figure be divided as required. Dem. By sim. tri. BL.AB=BQ2=BM2; BN. AB =BP2 and BO.AB=BT2; but the areas of squares are to each other in the duplicate ratio of the sides, hence dividing by AB which is common to all these equations BL: BN:: BM2: BP2, .-. the sqrs. described on BM and BP are in the given ratio, and in like manner BP2: BT2 and BT2: BA2 may be proved to be in the given ratio, by division of ratios Bin, PmF, TnG and ArC are in the ratio of BL, LN, NO and OA that is in the ratio of 2, 4, 6 and 8. Q. E. D. The tame, by Messrs. Archibald Hills, Milfield, and T. Black, Alnmouth. The required parts of the square being in the proportion of 1, 2, 3, 4, divide BA the side of the given sqr. in the same proportion, in L, N, O; upon BA describe a semicircle, draw perp. to BA the lines LQ, XR, OS meeting the circumf. in Q, R, S, join BQ, BR, BS and from B lay off BM, BP, BT, respectively equal to BQ, BR, BS,
