.. i A Right PROP. XVI. Theorem. Line (IF) which being drawn thro the Fig. Point (B), the Extremity of the Diameter (CB) is perpendicular thereto, falleth all of it without the Circle, and toucheth it in (B). Neither can there any right Line be drawn betwixt it and the Circle unto the Point of Contact (B), but it shall cut the Circle. 1. 20 Part 1. Let there be taken in the Line IBF any Point L, unto which from the Centre A draw the Line AL. Because in Triangle ABL, the Angle ABL is a right one, by the Hypothefis, AL B shall be acute (a). There-(a) Per Coroll. fore AL which is oppofite to the greater Angle B, will 5.p. 32. 1. 1. be greater than A B which is oppofite to the leffer Angle L (b). But A B reacheth only to the Circumference. (b) Per 19. Therefore AL shall reach beyond the Circumference; and confequently fall without the Circle. Which was the firft Part. 1.1. Part 2. Below BF, if it may be, let RB fall wholly without the Circle. Because FBA is a right Angle by the Hypothefis, RBA will be acute, and therefore A B is not perpendicular to BR. Therefore let there be drawn from the Centre A to BR the Perpendicular AO, which (c) will fall towards R, and cut the Circle (c) Per Coroll. in Q. Therefore AB which is oppofite to the greater 3. Prop. Angle AOB, is greater than AO, which is opposite to the leffer, to wit, the acute Angle OBA. But A B is equal to AQ: therefore AQ also is greater than A O, a Part than the Whole. Corollary. 1. 1.H Ence it appears again, that the Contact of a right Fig. 20. Line and a circular one, is only in one Point. 2. If from Centers taken in the same right Line infi- Fig. 17. nitely protracted, there be describ'd thro' B infinite Circles, as well lesser than the first BSC, as greater; they F2 32. they shall all touch the right Line IF in the same one Point B. 3. Circles therefore growing into an Amplitude greater than any given one, approach always, even unto Infinity, nearer and nearer to the Tangent, but are never join'd to it, otherwise than in one single Point of Contact; which thing altho' it be most evident, is yet truly admirable. 4. From these Things it is manifest, that every geometrical Line whatsoever is infinitely divisible. For let there be drawn from fome Point of the Diameter unto the Tangent the right Line AQ. Infinite Circles having Centres in the right Line B A infinitely produced; touch the right Line IF by Coroll. 2. of this, and one another by Coroll. p. 13. in one and the same Point B; and consequently are no where join'd either amongst themselves, or with the right Line IF, but in the Point B only. Therefore it is neceffary that they divide the right Line A Q into infinite Parts, that is, into Parts exceeding any Number affignable. 5. The Angle of Contingence or Contact LBQ, (that, to wit, which is contained under the Tangent and the Circumference) cannot be divided by any right Line. 6. Nevertheless by Circumferences touching the Line IF in the fame Point, it may be divided and diminished infinitely. And in this and the third Corollary lies hid the whole Mystery of Asymptotes, that is, of a right Line approaching unto an Hyperbola, together with it self infinitely produced, unto a Distance less than any given one, yet never concurring with it. F Rom PROP. XVII. Problem. a given Point (B) to draw a right Line which shall touch a given Circle (02). From A the Centre of the given Circle let there be drawn unto the Point B the right Line AB, cutting the Periphery in O. From the Centre A describe thro' B another Circle BC, and from O draw OC perpendicular to A B, which may meet the other Circle in C. Draw Draw CA meeting the Circle OQ in I. The right For fince the Sides BA, IA, are equal to the Sides CA, AO, and the Angle A contain'd betwixt the equal Sides in the Triangles I AB, OAC, is common to both; the Angles A OC, A I B are also (a) equal. Therefore(a) Per 41. 1: AIB is a right Angle. For AOC is a right one by the Construction. Therefore BI (b) toucheth the Circle (b) Per 16. in I. 1. 3. Scholium. BY the 31st of this Book, from the given Point O, a Fig: 27: Line touching a given Circle (BQ) may be well drawn thus: Let the right Line joining the given Point O and the Centre A be bisected in P. Then from the Centre P thro' A and O describe a Circle, meeting the given one in B. The right Line OB will touch the Circle. For A B being join'd, the Angle ABO in the Semicircle is a right one by Prop. 31. Therefore by Prop. 16. O B toucheth the Circle BQ. f I PROP. XVIII. Theorem. Fa right Line (CL) touch a Circle, a right Line Fig. 28; (AB) drawn from the Centre (A) unto the Point of Contact (B) is perpendicular to the Tangent. If it be denied, let some other right Line (as A F) be the Perpendicular from the Centre A. This will cut the Circle in O. Because therefore the Angle A FB is supposed to be a right one, ABF (c) must be acute. There-(c) Per Coral. fore A B (that is, AO) is greater than AF (d), a Part than the Whole; which is is abfurd. (d) Per 19. 1. 1. 70 Fig. 29... Fa EUCLID'S Elements. 1 PROP. XIX. Theorem. Lib. III. Line (BC) touch the Circle, and from the Point of Contact (A) there be rais'd (AI) perpendicular to the Tangent, the Centre will be in that Perpendicular. If you deny it, let the Centre be without AI in Z; and from it let there be drawn unto the Contact the (a) By the Line ZA. The Angle ZAC will be a right one (a), foregoing. and therefore equal to the Angle I AC, which by the Hypothefis is a right one; that is, the Part will be equal to the Whole, which is absurd. Fig. 30, 31, 32. Fig. 30. T PROP. XX. Theorem. to HE Angle at the Centre (BAC) is double when the Same Arch (BC) is the Base of the Angles. Here are three Cases. In the first Cafe the Sides BA, BF coincide. And then because AF, AC drawn from the Centre are equal, there will be in the Triangle (b) Per 5.1.1.Z, the Angles F and C equal (b). But BAC is equal (c) Per 32. to the two Angles F and C (c). Therefore BAC is double of F. 7. Σ. Fig. 31. 1 Fig. 32. In the second Cafe BA, CA fall within BF, CF; and then FAX being drawn, XAB by the first Case is double of X FB, and XAC double of X FC. Therefore the whole BAC is double of the whole BFC. In the third Cafe, BF cuts AC, and the Angle BAC is without the Triangle BFC. Here let FAL be drawn. By the first Case the whole LA Cis double of the whole LFC, and LAB taken away is double of LFB taken away. Therefore the remaining Angle BAC is also double of the remaining one B FC. 2. E. D. Fig. 53. [Corollary, Hence we gather that he Sides of every Triangle are to each other as the Sines of the Angles opposite to those Sides respectively. Let EFG be any Triangle; about which let a Circle be understood to be cir(d) Per 5.1.4.cumscrib'd (d), and from the Centre of the Circle, let there be let down the Perpendiculars AB, AC, AD, which will * bisect the Subtenses. Now as EF is to* Per 3.1.3. EG, fo EF (that is, EB) to EG (that is, ED.) But EB is the Sine of the Angle † BAE, that is, oft PerCoroll.2. half the Angle EAF, that is, of the whole Angle P. 3. 1. 3. EGF * opposite to the Side EF; and ED is the Sine* Per 201.3. of the Angle E AD, that is, of half the Angle EAG, that is, of the whole Angle EFG, which is opposite to the Side EG. Therefore EF is to EG, as the Sine of the Angle FGF, is to the Sine of the Angle EFG. Q.E.D. And from this one Proposition a great Part of Trigonometry is deduced. Which Thing will be worth our Observation. Coroll. (2.) From the former Corollary we learn to Fig. 86.1.1. measure the Distance of the Moon. For Astronomical Observations giving us the Angle of the Diurnal Paral. lax * BCA, we find out the Distance of the Moon by Coroll. 16. the following Proportion. As the Sine of the Anglet. 32. 1. 1. ACB is to the Sine of the Angle ABC; so is the Semidiameter of the Earth B A, unto the Moon's Distance AC. Q. E. I. Coroll. (3.) From the second Corollary we learn also to Fig. 54. measure the Distance of the Sun. For there being given by Astronomical Obfervations the Angle of the menstrual Parallax, (namely, that which is made when the Moon appears precisely bisected) or the Angle ZEO, and together with this Angle the Moon's Distance ZO; we find the Distance of the Sun by this Analogy. As the Sine of the Angle ZEO, is to the Sine of the Angle EOZ; which Sine is the Radius: So is ZO, the Moon's Distance, unto Z E the Distance of the Sun. Q.E.I.] T PROP. XXI. Theorem. HE Angles (BQC, BFC) which in a Circle Fig 33: Stand upon the Same Arch (BOC); or which are in the Same Segment (BQSC) are all equal among themselves. Let first the Segment BQSC be greater than a Semicircle. From the Centre A draw AB, AC. By the foregoing the Angle BAC at the Centre is double of each BQC, BFC. Therefore they all, BQ C, BFC, are equal (a). 2. E. D. F4 (a) Per axio. Then 6. 1 |