5. If equals be taken from unequals the remainders will be unequal.

6. Things which are double of the fame thing are equal to each other.

7. Things which are halves of the fame thing are equal to each other..

8. The whole is equal to all its parts taken together.

9. Magnitudes which coincide, or fill the fame space, are equal to each other.

REMARK S.

A PROPOSITION, is something which is either pro. posed to be done, or to be demonstrated.

A PROBLEM, is fomething which is propofed to be done.

A THEOREM, is something which is propofed to be demonftrated.

A LEMMA, is fomething which is previoufly demon, strated, in order to render what follows more easy.

A COROLLARY, is a consequent truth, gained from fome preceding truth, or demonftration.

A SCHOLIUM, is a remark or obfervation made upon "fomething going before it.

PROPOSITION I. PROBLEM.

UPON a given finite right line to describe an equilateral triangle.

E

Let AB be the given right line; it is required to describe an equilateral triangle upon it.

From the point A, at the diftance AB, defcribe the circle BCD (Pof. 3.)

And from the point B, at the distance BA, describe the circle ACE (Pof. 3.)

... Then, because the two circles pass through each other's centres, they will cut each other.

And, if the right lines CA, CB be drawn from the point of interfection C, ABC will be the equilateral triangle re-/ quired.

For, fince A is the centre of the circle BCD, ac is equal to AB (Def. 13.) ·

And, because B is the centre of the circle ACE, BC is alfo equal to AB (Def. 13.)

But things which are equal to the fame thing are equal to each other (Ax. 1); therefore AC is equal to CB.

And, fince AC, CB are equal to each other, as well as to AB, the triangle, ABC is equilateral; and it is defcribed upon the right line AB, as was to be done."

PROP. II. PROBLÈM.

From a given point to draw a right line equal to a given finite right line.

E

B

Let A be the given point, and BC the given right line; it is required to draw a right line from the point A, that fhall be equal to BC.

Join the points A, B, (Pos. 1.); and upon BA defcribe the equilateral triangle BAD (Prop. 1.),

From the point B, at the distance BC, describe the circle CEF (Pof. 3.) cutting DB produced in F.

And from the point D, at the distance DF, defcribe the circle FHG (Pof. 3.); then, if DA be produced to G, AG will be equal to BC, as was required.

For, fince B is the centre of the circle CEF, BC is equal to BF (Def. 13.)

And, because D is the centre of the circle FHG, DG is equal to DF (Def. 13.)

But the part DA is alfo equal to the part DB (Def. 16.), whence the remainder AG will be equal to the remainder BF (Ax. 3.)

And fince AG, BC have been each proved to be equal to BF, AG will alfo be equal to BC (Ax. 1.)

A right line AG, has, therefore, been drawn from the point A, equal to the right line BC, as was to be done.

SCHOLIUM. When the point A is at one of the extremities B, of the given line BC, any right line, drawn from that point to the circumference of the circle CEF, will be the one required.

From the greater of two given right lines, to cut off a part equal to the lefs.

好

Let AB and c be the two given right lines; it is required to cut off a part from AB, the greater, equal to c

the lefs.

From the point A draw the right line AD equal to c (Prop. 2.); and from the centre A, at the distance AD, defcribe the circle DEF (Pof. 5.) cutting AB in E, and AE will be equal to c as was required.

For, fince A is the centre of the circle EDF, AE will be equal to AD (Def. 13.)

But c is equal to AD, by conftruction; therefore AE will alfo be equal to c (Ax. 1.)

Whence, from AB, the greater of the two given lines, there has been taken a part equal to c the lefs, which was to be done.

SCHOLIUM. When the two given lines are fo fituate, that one of the extremities of c falls in the point A, the former part of the construction becomes unneceffary.

PROP.

attention have been bestowed upon the work; and that nothing which was judged effential to the fcience, or useful in facilitating its attainment, has been omitted. The acknowledged intricacy of fome propofitions in the fifth and fixth books, made it neceffary to abridge that part of the fubject more confiderably than the former; but it is conceived that what is here given will be fully fufficient to anfwer all the purposes of the learner.

To avoid critical objections were a vain endeavour they may be made against every fyftem of Geometry now extant; and to EUCLID as well as to other writers. Of this abundant proofs are given by the Commentators; and in the Notes at the end of the prefent work, where many things of this kind are pointed out which have hitherto escaped notice. Thefe were added chiefly for the information of young ftudents, and ought to be carefully confulted by those who wish to obtain a juft idea of the science, and the principles upon which it is founded.