concerning the same may vanish ; as that the Square of the Hypothenuse of a right-angled Triangle, is equal to the two Squares of the other Sides. A Problem is something proposed to be done; as to make a Circle pass thro' three given Points. A Lemma is a Proposition laid down only for demonstrating some following Proposition more eafily.

A Corollary is a Consequence deduced from a foregoing Proposition.

A Scholium is a Critical Exposition upon something said before.

Poftulates or Petitions.

1. Grant that a right Line may be drawn from any Point to any other Point.

2. That a finite right Line may be drawn

out at pleasure.

3. That a Circle may be described about any Centre with any Distance.

Axioms.

1. Things equal to the same third Thing, are also equal the one to the other.

As, if A = B = C; then is A=C: or therefore all the Quantities A, B, C, are equal the one to the other.

Note. When you find several Quantities connected together after this manner, the first Quantity, or any of the intermediate ones, by virtue of this Axiom, is equal to the last: in which cafe, for brevity's fake, we usually omit citing this Axiom, notwithstanding the Force of the Consequence depends upon it.

2. If to equal things you add equal things, the wholes shall be equal. As if A = B, and C=D; then shall A+C=B+D.

3. If

3. If from equal things you take away equa things, the things remaining will be equal. As if A+C=B+D, and A = B; then shall C=D..

4. If to unequal things you add equal things, the wholes will be unequal. As if A Band C=D; then shall A+ CB+D.

5. If from unequal things you take away equal things, the remainders will be unequal. As if A+CB+D, and C=D: then shall AD.

6. Things which are the double of the same third thing, or of equal things, are equal one to the other. Understand the fame of things that are the triple, quadruple, &c. of the same thing. As if A = 2 B, and C=2B: then shall A

С.

7. Things which are the half of the one and the fame thing, or of equal things, are equal the one to the other; conceive the same of things that are the one third, one fourth of the fame thing. As if A = + B, and C =+B: then shall A = C.

8. Things which agree together, or coincide, are equal the one to the other.

The Converse of this Axiom is true in right Lines and Angles, but not in Figures, unless they be similar.

Moreover, Magnitudes are said to agree, or coincide, when the parts of the one being apply'd to the parts of the other, they fill up an equal, or the same Space.

9. Every Whole is greater than its Part. 10. Two right Lines cannot have one and the same Segment (or Part) common to them both.

A

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 A D, CB, makes the internal Angles on

the fame Side BAD+ ABC less than two right Angles: those two right Lines produced, shall meet on that Side, where the Angles are less 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 shall be equal to the excess of the unequal things before the Addition. As if A = B, and C-D: then shall A+C-B-D=C-D.

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

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

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

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 first, the double of that which is taken away from the second, the remainder of the first shall. be the double of the remainder of the second. As if A = 2B, and C=2D: then shall A-C =2B-2D.

The Citations are thus to be understood : When you meet with two Numbers in the Margin, the first shews the Proposition, the fecond the Book; as by 4.1. you are to understand the fourth Propofition of the first Book : and so of the rest. Moreover, ax. denotes Axiom, post. Postulate, def. Definition, Sch. Schotium, and cor. Corollary.

PROPOSITION I.

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

:

a 3 post.

bi poft.

C15 def.

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.

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

PROP. II.

From a given Point A, to draw a right Line

AG equal to a right Line given (BC).

CI. I.

About the Center C, with the Distance CB, describe the Circle CBE. Join AC, upon a 3 poft. which raise the equilateral Triangle ADC. bi poft. Produce d DC to E, about the Center D, with d 21 the Distance DE, describe the Circle DEH, e 2 poft. and let DA be produced to the Point Ginf 15 def. the Circumference thereof. Then is AG=CB. & conft.

poft.

For DG £ = DE, and DA = DC. Where-h 3 ax.

k

i 15 def. fore AG = CE = BC = AG. Which was k í ax. to be done.

The Position of the Point A within or without the Line BC varies the Cases, but the Construction and Demonstration are every where the fame.

SCHOLIUM.

The Line AG might be taken between the Points of a Pair of Compasses, but no Postu