For let Ab F be the nearest ray to ABF; and upon Ab draw the perpendiculars Cd, Bd; and Ce, By upon b F; and the right angled triangles B&b, BDC will be fimilar, as appears by taking the common angle DBb from the right angles CBb, DBJ. And for the fame reason the right angled triangles Bub, BEC will be fimilar; and confequently the whole figures Badb, BEDC will alfo be fimilar. It is alfo to be obferved that CD is to CE and Cd to Ce in the given ratio of the fine of incidence to the fine of refraction; and disjointly Dd is to Ee in the fame ratio. For the fame line CDd may be confidered as perpendicular to both rays AB, Ab when the angle BAb is vanishing. Hence because the triangles a Art. 20+ BF, E Fe are fimilar; and alfo BA, DAd; the ratio of BF to EF or of By to Ee, which is compounded of the ratios В to Вd, Bd to Dd, Dd to Ee, is alfo compounded of these ratios refpectively the fame as the former, BE to BD, BA to DA, CD to CE. 2. E. D. The pofition of the refracted ray BF, which is fuppofed to be given in this and the following propofitions, will be determined hereafter by lemma 4.

413. Corol. 1. When the incident rays are parallel, the ratio of BF to EF is compounded of the direct ratio of the fines of incidence and refraction, and of the inverse ratio of the cofines; because the ratio of BA to DA is a ratio of equality when A is remote.

414. Corol. 2. The ratio of the perpendicular fubtenfes Dd, Ee, of the fmall angles at A and F, is invariable; being the fame as of the fine of incidence CD to the fine of refraction CE.

LEMMA. I.

415. The ratio of the tangents, CT, CV, of any two angles, CBD, Fig. 413CBE, is compounded of the ratio of their fines, CD, CE, taken directly, and of their cofines, BD, BE, taken inverfely.

For the right angled triangles BCT, BDC are equiangular, and so are the right angled triangles BCV, BEC. Therefore the ratio of CT to CV, which is compounded of CT to CB and of CB to CV, or of CD to DB and of EB to EC, is the fame as the ratio of the rectangle under CD, EB to the rectangle under DB, EC, which is compounded of the ratio of CD to CE and of EB to DB, that is of the fines directly and cofines b Euc. VI. 23 inverfely. 2. E. D.

PROPOSITION IV.

416. Let AB and Bf be an incident and a refracted ray given in pofi- Fig. 414 to tion. From C the center of curvity at B, or of the refracting circle, draw 417. CE perpendicular to the refracted ray B fproduced; and let Bfbe to BE as the tangent of the angle of incidence to the difference of the tangents of incidence and refraction; and let Bf be placed forward, that is according to the courfe

a Art. 413.

b Art. 415.

C

of the refracted ray, if the furface of the denfer medium be convex, otherwise backwards; and f will be the focus of a flender pencil of rays that came parallel to AB upon the fmalleft arch at B.

For the ratio of Bf to Ef, being compounded of the ratio of the fines of the angles of incidence and refraction directly and of their cofines inverfely, is the fame as the ratio of the tangents of thofe angles"; and disjointly Bf is to BE as the tangent of incidence to the difference of the tangents of incidence and refraction. The reason of the rule for the position of Bf is this, that the rays going forward in both cafes, will converge. in one cafe and diverge in the other. 2. E. D.

417. Corol. 1. Hence Bf is to Ef as the tangent of incidence to the tangent of refraction.

418. Corol. 2. Let a be the focus of rays that came parallel to ƒ B; then we have Bf to Da as BE to BD, that is, the continuations of the cofines BE, BD to the focuses of rays that came parallel to them, are in the fame ratio as the cofines themselves. For Bf is to Ef as the tangent of incidence to the tangent of refraction, and Ba to Da in the fame ratio inverted. Therefore Bf: Ef:: Da: Ba, and disjointly Bf: BE:: Da: DB, and alternately Bf: Da:: BE:BD.

419. Corol. 3. The focus f may alfo be found by drawing lines in this manner. Draw GE perpendicular to the refracted ray Ef; EG perpendicular to the radius BC, and Gf parallel to the incident rays, and it will cut the refracted rays in their focus f. For let EG cut AB in H, and Gf being parallel to the bafe BH of the triangle BEH, we have Bf: BE:: Euc. VI. 2. HG: HE, the fame proportion as in the propofition: for GH and GE are tangents of the angles, GBH, GBE, of incidence and refraction.

d Art. 224. e Art. 204.

Fig 418,419.

Art. 204.

420. Corol. 4. Hence it appears that while the angle of incidence continually increafes, the focal distance Bf continually decreases, till it be reduced to nothing when the angle of refraction becomes a right one; or elte till it be equal to the cofine of refraction when the angle of incidence is a right one. Confequently the focal distance is the longest when the angle of incidence is the leaft; and then this propofition degenerates to the fecond propofition of the third chapter d. For the tangents of very fmall angles are in the fame ratio as their fines or arches.

LEMMA II.

421. The leaft increment of an angle of incidence, is to the contemporary increment of the angle of refraction, as the tangent of the angle of incidence, to the tangent of the angle of refraction.

Let two rays AB, a B, containing a very small angle ABa, be refracted at Balong the lines BE, Be by a plane or by any curve-furface. From any point C, of the line BC perpendicular to that furface, draw CD d cutting the incident rays (produced) at right angles f in D and d; and

likewife CEe cutting the refracted rays (produced) at right angles in E and e. Then becaufe CD is to CE and Cd to Ce in the fame ratio of the fines, disjointly we have Dd to Ee as CD to CE. Now the ratio of the fmall angles (A Ba or) D Bd and EBe, which are the contemporary increments or decrements of the angles of incidence and refraction, being compounded of the ratio Dd to Ee and of BE to BD, that is of CD to a Art. 222. CE and of BE to BD, is the fame as the ratio of the tangents of incidence and refraction'. 2. E. D.

422. Corol. Hence if the angles of incidence and refraction of either of the rays ABe, a Be be invariable, while thofe of the other ray are varied a little, their small increments or decrements will be always to each other in an invariable ratio.

PROPOSITION V.

b Art. 415

423. Let a ray AB falling with any obliquity upon a refracting curve Fig. 414 to at B, be refracted along BF given in pofition; and let Ba be the focal di- 417. Stance of rays coming parallel to FB; and Bf the focal diftance of other rays coming parallel to AB; then fuppofing any point A to be a given focus of incident rays, fay as Aa to a B fo Bf to fF, and place fF the fame way with respect to fB as a A lyes with respect to a B; and F will be the focus of the refacted rays.

For let AGF be the neareft ray to ABF; and joining a G and fG, a Fig. 420. ray a G after refraction at G will go along a line G H parallel to BF, by the fuppofition; and likewise a ray fG will be refracted through G along GI parallel to BA. But the angle AG a is to the angle FGH or GFfin a certain given ratio; and in like manner the angle a AG or AGI is to c Art. 422. fGF in the fame given ratio. Therefore alternately the angle AGa: ang. a AG:: ang. GFf: ang. fGF; and the fines of these small angles are in the fame proportion; and confequently in the triangles AGa, GFf, the d Art. 220. fides oppofite to those angles are also in the fame proportion; that is Aa e Art. 221. :aG::Gf:fF, or Aa: a B:: Bf:fF. Let A approach towards a, and f Art. 204. when it coincides with a, the line fF will become infinite; and therefore when A paffes to the other fide of a, the point F will also pass from an infinite distance to the other fide off. 2.E. D.

424. Corol. 1. When the ray ABF is given in pofition the line ƒF is reciprocally as Aa.

425. Corol. 2. The fame words applyed to the 421ft figure are a de- Fig. 421. monstration of the fame proportion for the focus of reflected rays. For now the angle AGa is to FGH or GFf in a ratio of equality, and so is 8 Art. 8. the angle a AG or AGI to the angle fGF; and confequently the triangles AGa, GFf are equiangular; and therefore Aa: a G or a B:: Gfor Bf:fF. Which is another demonftration of the first of these propofitions; and the points a,f which are the focuses of parallel rays to FB h Art. 402.

and