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The first six books of
the Elements of Euclid by Oliver Byrne
1.

A point is that which has no parts.

2.

A line is length with­out breadth.

3.

The ex­trem­i­ties of a line are points.

4.

A straight or right line is that which lies even­ly be­tween its ex­trem­i­ties.

5.

A sur­face is that which has length and breadth only.

6.

The ex­trem­i­ties of a sur­face are lines.

7.

A plane sur­face is that which lies even­ly be­tween its ex­trem­i­ties.

8.

A plane an­gle is the in­cli­na­tion of two lines to one an­oth­er, in a plane, which meet to­geth­er, but are not in the same di­rec­tion.

9.

A plane rec­ti­lin­ear an­gle is the in­cli­na­tion of two straight lines to one an­oth­er, which meet to­geth­er, but are not in the same straight line.

10.

When one straight line stand­ing on an­oth­er straight line makes the ad­ja­cent an­gles equal, each of these an­gles is called a right an­gle, and each of these lines is said to be per­pen­dic­ular to the oth­er.

11.

An ob­tuse an­gle is an an­gle great­er than a right angle.

12.

An acute an­gle is less than a right an­gle.

13.

A term or bound­ary is the ex­trem­ity of any thing.

14.

A fig­ure is a sur­face en­closed on all sides by a line or lines.

15.

A cir­cle is a plane fig­ure, bound­ed by one con­tin­ued line, called its cir­cum­fer­ence or pe­riph­ery; and hav­ing a cer­tain point with­in it, from which all straight lines drawn to its cir­cum­fer­ence are equal.

16.

This point (from which the equal lines are drawn) is called the cen­tre of the cir­cle.

17.

A di­am­e­ter of a cir­cle is a straight line drawn through the cen­tre, ter­mi­nat­ed both ways in the cir­cum­fer­ence.

18.

A sem­i­cir­cle is the fig­ure con­tained by the di­am­e­ter, and the part of the cir­cle cut off by the di­am­e­ter.

19.

A seg­ment of a cir­cle is a fig­ure con­tained by a straight line, and the part of the cir­cum­fer­ence which it cuts off.

20.

A fig­ure con­tained by straight lines only, is called a rec­ti­lin­ear fig­ure.

21.

A tri­an­gle is a rec­ti­lin­ear fig­ure in­clud­ed by three sides.

22.

A quad­ri­lat­eral fig­ure is one which is bound­ed by four sides. The straight lines and con­nect­ing the ver­ti­ces of the op­po­site an­gles of a quad­ri­lat­eral fig­ure, are called its di­ag­o­nal.

23.

A pol­y­gon is a rec­ti­lin­ear fig­ure bound­ed by more than four sides.

24.

A tri­an­gle whose three sides are equal, is said to be equi­lat­er­al.

25.

A tri­an­gle which has only two sides equal is called an isos­ce­les tri­an­gle.

26.

A sca­lene tri­an­gle is one which has no two sides equal.

27.

A right an­gled tri­an­gle is that which has a right an­gle.

28.

An ob­tuse an­gled tri­an­gle is that which has an ob­tuse an­gle.

29.

An acute an­gled tri­an­gle is that which has three acute an­gles.

30.

Of four­sid­ed fig­ures, a square is that which has all its sides equal, and all its an­gles right an­gles.

31.

A rhom­bus is that which has all its sides equal, but its an­gles are not right an­gles.

32.

An ob­long is that which has all its an­gles right an­gles, but has not all its sides equal.

33.

A rhom­boid is that which has its op­po­site sides equal to one an­oth­er, but all its sides are not equal, nor its an­gles right an­gles.

34.

All oth­er quad­ri­lat­eral fig­ures are called tra­pe­zi­ums.

35.

Par­al­lel straight lines are such as are in the same plane, and which be­ing pro­duced con­tin­u­ally in both di­rec­tions, would nev­er meet.
Postulates
I.

Let it be grant­ed that a straight line may be drawn from any one point to any oth­er point.

II.

Let it be grant­ed that a fi­nite straight line may be pro­duced to any length in a straight line.

III.

Let it be grant­ed that a cir­cle may be de­scribed with any cen­tre at any dis­tance from that cen­tre.
I.

Mag­ni­tudes which are equal to the same are equal to each oth­er.

II.

If equals be add­ed to equals the sums will be equal.

III.

If equals be tak­en away from equals the re­main­ders will be equal.

IV.

If equals be add­ed to une­quals the sums will be une­qual.

V.

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

VI.

The dou­bles of the same or equal mag­ni­tudes are equal.

VII.

The halves of the same or equal mag­ni­tudes are equal.

VIII.

The mag­ni­tudes which co­in­cide with one an­oth­er, or ex­act­ly fill the same space, are equal.

IX.

The whole is great­er than its part.

X.

Two straight lines can­not in­clude a space.

XI.

All right an­gles are equal.

XII.

If two straight lines () meet a third straight line () so as to make the two in­te­ri­or an­gles ( and ) on the same side less than two straight an­gles, these two straight lines will meet if they be pro­duced on that side on which the an­gles are less than two right an­gles.

NN a giv­en fi­nite straight line () to de­scribe an equi­lat­eral triangle.

P

Describe and (post. III) ;

draw and (post. I).

Then will be equilateral.

For = (def. 15) ;
and = (def. 15) ;
∴ = (ax. I) ;

and therefore is the equilateral triangle required.

Q. E. D.

PROM a giv­en point (), to draw a straight line equal to a given straight line ().

P

Draw (post. I), describe (pr. I.I),

produce (post. II),

describe (post. III), and (post. III) ;

produce (post. II), then is the line required.

For = (def. 15),
and = (const.),
∴ = (ax. III), ————————
but (def. 15) = =  ;
∴ drawn from the given point
is equal to the given line .

Q. E. D.

PROM the greater (), of two given straight lines, to cut off a part equal to the less ().

P

Draw = (pr. 2.) ;

describe (post. 3.),

then = .

For = (def. 15.),
and = (const.) ;
∴ = (ax. I.).

Q. E. D.

Oliver Byrne, The First Six Books of the Elements of Euclid, 1847.