11. Two right Lines meeting in the fame Point, if they be both produced, will neceffarily cut one another in that Point.

12. All right Angles are equal to one another.

13. If a right Line BA, falling on two right Lines AD, CB, makes the internal Angles on

the fame Side B AD+ABC lefs than two right Angles: thofe two right Lines produced, fhall meet on that Side, where the Angles are lefs than two right Angles.

14. Two right Lines do not contain a Space. 15. If to equal things you add unequal things, the excess of the wholes fhall be equal to the excefs of the unequal things before the Addition. As if AB, and CD: then shall A+C-B-D=C-D.

16. If to unequal things equal things be added, the excess of the whole fhall be equal to the excess of the unequal things before the Addition. If A =B, and CD; then shall A+ C-B-D be A-B.

17. If from equal things unequal things be taken away, the excefs of the 'remainders fhall be equal to the excess of the things taken away. If AB, and C=D; then fhall A-C-B +D be+D-C.

18. If equal things be taken from unequal ones, the excefs of the remainders fhall be equal to the excefs of the wholes. As if AB, and C=D; then fhall A-C+D-B be A-B,

19. E

19. Every Whole is equal to all its Parts taken together.

20. If one whole thing be the double of another, and that which is taken away from the firft, the double of that which is taken away from the fecond, the remainder of the firft fhall. be the double of the remainder of the fecond. As if A 2B, and C 2D: then fhall A-C =2B2D.

The Citations are thus to be understood: When you meet with two Numbers in the Margin, the firft fhews the Propofition, the fecond the Book; as by 4. 1. you are to underftand the fourth Propofition of the first Book: and fo of the reft. Moreover, ax. denotes Axiom, poft. Poftulate, def. Definition, fch. Scholium, and cor. Corollary.

PROPOSITION I.

Upon a given finite right Line (AB) to defcribe an equilateral Triangle (ABC).

b

*About the Centers A and B, with the common distance AB or BA, describe two Circles cutting each other in the Point C; from which draw two right Lines, CA, CB: then is AC =AB BC AC. Wherefore the Triangle ACB is an equilateral one. Which was to be done.

SCHO

SCHOLIUM.

After the fame manner may an Ifofceles Triangle be described upon the Line AB, if the Distances or Intervals of the equal Circles be taken greater or lefs than the Line AB.

PROP. II.

From a given Point A, to draw a right Line AG equal to a right Line given (BC).

a

с

b

bi

C I. I.

d 2 poft.

About the Center C, with the Distance CB, defcribe the Circle CBE. Join AC, upon a 3 poft. which raise the equilateral Triangle ADC. poft. Produced DC to E, about the Center D, with the Distance DE, defcribe the Circle DEH, e 2 poft. and let DA be produced to the Point G inf 15 def. the Circumference thereof. Then is AGCB. g conft. For DG DE, and DA DC. Where- 3 ax. fore AGCE BC AG. Which was k 14x. i is def to be done.

=

The Pofition of the Point A within or without the Line BC varies the Cafes, but the Conftruction and Demonftration are every where the fame.

SCHOLIUM.

The Line AG might be taken between the Points of a Pair of Compaffes, but no Poftu

late

h

late of Euclid allows the doing of this, as Proclus well obferves.

PROP. III.

Two right Lines (A and BC) being given; from the greater (BC) to take away a right Line (BE), equal to the leffer (A.)

=

From the Point B draw the right Line BD A; then a Circle defcribed about the Center B, with the Distance BD, fhall cut off BEBDA BE. Which was to be

done.

PROP. IV.

1

If two Triangles (BAC, bac) have two Sides of the one (BA, AC) equal to two Sides of the other (ba, ac) each to its correfpondent Side (that is, BA =ba, and AC=ac) and have one Angle (A) equal to one Angle (a), contained under the equal right Lines: they fhall have the Base (BC) equal to the Bafe (bc), and the Triangle (BAC) shall be e

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B

qua

qual to the Triangle (bac);, and the remaining Angles (B, C) fhall be equal to the remaining Angles (b, c) each to each, under which the equal Sides are fubtended.

If the Point a be applied to the Point A, and the right Line ab placed upon the right Line AB, the Point b fhall fall upon B, because ab AB alfo the right Line ac fhall fall upon AC, because the Angle Aa; moreover, a hyp. the Point ofhall fall upon the Point C, because AC ac. Therefore the right Lines bc, BC, because they have the fame Bounds, fhall agree bor coincide, and fo confequently are equal. b 8 ax. Wherefore the Triangles BAC, bac, and the Angles B, b, as alfo the Angles C, c, do agree, or coincide, and are equal. Which was to be demonStrated.

B

D

e

A

PROP. V.

C

The Angles (ABC, ACB) at the Bafe of an Ifofceles Triangle are equal to each other. And if the equal right Lines AB, AC, be continued out, the Angles (CBD,BCE) under the Bafe, are equal to each other.

a

4 1.

Take AE AD, and a 3. 1. join C, D, and, B, E.bi poft. Then in the Trianglese hyp. ACD, ABE, becaufe AB is = AC, and A Edd conft. AD, and the Angle A is common: therefore fhall the Angle ABE be ACD, and the e. Angle AEB ADC, and the Bafe BEDC. Allo EC is DB. Whence in the Triangles f 3 ax. BCE, BDC, the Angle ECB fhall be = DBC, (which is the latter part of the Propofition to be demonftrated) and confequently the Angle EBC= DCB;

e

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