Apollonius <Pergaeus>; Lawson, John - The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus : with a supplement to which is now added, a second supplement, being Mons. Fermat's Treatise on spherical tangencies

[51.] PROBLEM X.

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[52.] PROBLEM XI.

[53.] PROBLEM XII.

[54.] PROBLEM XIII.

[55.] PROBLEM XIV.

[56.] PROBLEM XV.

[57.] Synopſis of the PROBLEMS.

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[58.] THE TWO BOOKS OF APOLLONIUS PERGÆUS, CONCERNING DETERMINATE SECTION, As they have been Reſtored by WILLEBRORDUS SNELLIUS. By JOHN LAWSON, B. D. Rector of Swanſcombe, Kent. TO WHICH ARE ADDED, THE SAME TWO BOOKS, BY WILLIAM WALES, BEING AN ENTIRE NEW WORK. LONDON: Printed by G. BIGG, Succeſſor to D. LEACH. And ſold by B. White, in Fleet-Street; L. Davis, in Holborne; J. Nourse, in the Strand; and T. Payne, near the Mews-Gate. MDCC LXXII.

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[59.] ADVERTISEMENT.

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[60.] EXTRACT from PAPPUS's Preface to his Seventh Book in Dr. HALLEY's Tranſlation. DE SECTIONE DETERMINATA II.

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[61.] THE PREFACE.

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[62.] PROBLEMS CONCERNING DETERMINATE SECTION. PROBLEM I.

[63.] LEMMA I.

[64.] LEMMA II.

[65.] LEMMA III.

[66.] PROBLEM II.

[67.] LEMMA IV.

[68.] LEMMA V.

[69.] PROBLEM III.

[70.] PROBLEM IV.

[71.] DETERMINATE SECTION. BOOK I. PROBLEM I. (Fig. 1.)

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[72.] PROBLEM II. (Fig. 2 and 3.)

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[73.] PROBLEM III. (Fig. 4. and 5.)

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[74.] PROBLEM IV. (Fig. 6. 7. and 8.)

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[75.] PROBLEM V. (Fig. 9. 10. 11. 12. 13. 14. 15. 16.)

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[76.] PROBLEM VI. (Fig. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.)

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[77.] THE END OF BOOK I.

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[78.] DETERMINATE SECTION. BOOK II. LEMMA I.

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[79.] LEMMA II.

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[80.] LEMMA III.

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109[32] CU as IO is to BO; and ſo, by compound ratio, the rectangle AO, UO
is to the rectangle BO, CU as the rectangle EO, IO is to the rectangle
BO, CE; by permutation, the rectangle AO, UO is to the rectangle EO,
IO, as the rectangle BO, CU is to the rectangle BO, CE; or (Eu. V. 15.)
as CU is to CE; that is, by conſtruction as R to S.

Q. E. D.

Scholium. In enumerating the ſeveral Caſes of this Problem I ſhall en-
deavour to follow the method which I conceive Apollonius did: and there-
fore, notwithſtanding the preceding Analyſis and Conſtruction are general
for the whole, divide it into three Problems, each Problem into three Epi-
tagmas, or general Caſes, and theſe again into their ſeveral particular ones.

PROBLEM I. (Fig. 32 to 45.)

Here O is ſought between the two mean points of the four given ones:
and the three Epitagmas are, firſt, when A and U, the points which bound
the ſegments containing the antecedent rectangle, are one an extreme, and
the other an alternate mean; ſecondly, when thoſe points are one an ex-
treme and the other an adjacent mean; thirdly, when they are both means,
or both extremes.

Epitagma I, Conſiſts of eight Caſes, viz. when the order of the given
points is A, I, U, E; U, E, A, I; A, E, U, I; or U, I, A, E, and the
given ratio of a leſs to a greater, and four others wherein the order of the
points is the ſame as in thoſe, but the ratio of R to S, the ratio of a greater
to a leſs.

Case I. Let the order of the given points be A, I, U, E, and the given
ratio of a leſs to a greater; and the Conſtruction will be as in Fig. 32, where
B is made to fall beyond A, with reſpect to I, and C beyond U with re-
ſpect to E, and DH is drawn through F, the center of the circle on BC.
That O, when this conſtruction is uſed, will fall between I and U is
plain, becauſe CO is to CU as IB is to BO; and therefore if CU be
greater than CO, BO will be greater than IB, and if leſs, leſs; but this,
it is manifeſt, cannot be the Caſe if O falls either beyond I or U, and
therefore it falls between them.

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