Notes on the First Two Books of Euclid

Designed for Candidates for Responsions.

Notes on the Definitions

A Point. It must not be supposed that a Point has only negative qualities, in which case it would be identical with absolute nothing. It has the positive quality of position, and it is this which enables us to distinguish one Point from another.

A Plane Angle. It should be observed that an angle is not the point where two lines intersect; it is not the lines themselves; it is not the space between them; but it is the “inclination,” (or “bending,”) of the two lines to each other.

A Triangle. A Triangle may be said to consist of 6 parts, viz., 3 sides and 3 angles. In order to prove one Triangle equal in all respects to another, it must be first given, (or proved) that 3 of these 6 parts in the one are equal to the corresponding 3 in the other. The various cases are:—

Prop. IV. Two sides and the included angle of the one equal to two sides and the included angle of the other.

Prop. VII. The three sides of the one equal to the three sides of the other.

Prop. XXVI. Two angles and one side of the one equal to two angles and one side of the other.

Additional Definitions

I.

A Postulate is something to be done, for which no proof is given. (From postulatum, because it is demanded that the reader should grant it to be possible.)

II.

An Axiom is something to be believed, for which no proof is given. (From ἀξίωμα, because it is thought worthy of belief.)

III.

A Proposition is something either to be done, or to be believed, for which a proof is given. (From propositum, because it is put before the reader.) Propositions therefore are of 2 kinds, Problems, and Theorems.

IV.

A Problem is something to be done, for which a proof is given. (From πρόβλημα, for the same reason as the last.)

V.

A Theorem is something to be believed, for which a proof is given. (From ϑεώρημα, because it is to be considered, as to its being true or false.)

VI.

The Enunciation of a Proposition states (1) what is given, (2) what is required to be done or proved true: these two parts of the Enunciation are called the Data, and the Quæsita.

VII.

The Hypothesis of a Proposition is the same as its Data: it is a word used only with reference to Theorems. (From ὑπόϑεσις, because it is supposed to be true.)

VIII.

A Corollary is something proved in the course of Proposition, which it was not the object of the Proposition to prove. (From corolla, because it is a sort of ornament, or garland, of the Proposition.)

IX.

Two Propositions are said to be converse to each other when that which is given in the first is to be done or believed in the second, and that which is to be done or believed in the first is given in the second. For example, the Propositions, “if 2 sides of a triangle be equal, the opposite angles are equal,” and “if 2 angles of a triangle be equal, the opposite sides are equal,” are converse to each other.

X.

“A fortiori,” (or “much more then,”) is a phrase used when the conclusion has a stronger claim to be believed than any of the facts from which it is proved. For example, “3 is greater than 2, and 4 is greater than 3; much more then is 4 greater than 2.”

XI.

Reductio ad absurdum is the name of a form of argument, in which something is proved true, by showing that, if it were not true, an absurdity, (or impossibility,) would follow. For example, Prop. XXVII. “If a straight line, falling on 2 other straight lines, make the alternate angles equal to each other; these 2 straight lines shall be parallel,” is proved true, by showing that, if they were not parallel, an absurdity would follow.

The following scheme may assist the reader in understanding the first five Definitions given above.

Things set before us in Geometry may be of 2 kinds, viz.:
$\overset{⏞}{}$
(without proofs)		(with proofs)
		i. e. Propositions.
$\overset{⏞}{}$		$\overset{⏞}{}$
(to be done)	(to be believed)	(to be done)	(to be believed)
i. e. Postulates.	i. e. Axioms.	i. e. Problems.	i. e. Theorems.

A Proposition may be divided into the following parts:—

I. General Enunciation.
(1) Data.
(2) Quæsita.
II. Particular Enunciation.
(1) Data.
(2) Quæsita.
III. Construction.
IV. Proof.
V. Particular Conclusion.
VI. General Conclusion. (Only found in Theorems.)

Take, for example, Prop. VI of Book I.

I. General Enunciation.
(1) Data.
If two angles of a triangle be equal to each other,
(2) Quæsita.
the sides also which subtend the equal angles, shall be equal to each other.
II. Particular Enunciation.
(1) Data.
Let $A B C$ be a ▵ having $∠ A B C = ∠ A C B$ ,
(2) Quæsita.
then side $A B$ shall = side $A C$ .
III. Construction.
For, if $A B \neq A C$ , one of them is > the other;
let $A B$ be $> A C$ ;
from $B A$ cut off $B D = A C$ , and join $D C$ ,
IV. Proof.
Then, in ▵s $D B C$ , $A B C$ ,
∵ $D B = A C$ , and $B C$ is common, and $∠ D B C = ∠ A B C$ ;
∴ base $D C$ = base $A B$ ,
and $▵ D B C = ▵ A B C$ ,
the < = the >, which is absurd;
∴ $A B$ is not $\neq A C$ ;
V. Particular Conclusion.
that is, $A B = A C$ .
VI. General Conclusion.
Therefore, if two triangles, &c. Q. E. D.

Notes on the Propositions

When two Propositions are converse to each other, the second has generally no “construction,” and is proved by a “reductio ad absurdum.”

Book I. Prop. XVI. The second part may be proved thus:—

Geometrical drawing of a triangle ABC. The side BC is produced to D, the side AC is produced to G. H and K are constructed as explained below.

Bisect $B C$ in H, join $A H$ , and produce it to K, making $H K = A H$ , and join $C K$ ;

∵ $A H = H K$ , and $B H = H C$ , and $∠ A H B = ∠ K H C$ , (being vertical angles);

∴ $C K = A B$ , and $∠ K C H = ∠ A B H$ , i. e. $∠ A B C$ ;

but $∠ G C B > ∠ K C B$ ;

∴ $∠ G C B > ∠ A B C$ ;

and $∠ A C D = ∠ G C B$ , (being vertical angles);

∴ $∠ A C B > ∠ A B C$ . Q. E. D.

Book I. Prop. XXXV. Here the taking away of the triangles from the trapezium is meant to be done thus:—

From the trapezium $A B C F$ take the triangle $F D C$ , and observe that there remains the parallelogram $A B C D$ ; then replace this triangle, so as to make the trapezium complete again; then take away from it the triangle $E A B$ , and observe that there remains the parallelogram $E B C F$ . Since, in the two subtractions, equal triangles are taken away, the remainders are equal.

Book II. Prop. VII. It should be observed, that on the space $C G K B$ there are supposed to be two squares, one lying over the other. One of these is supposed to be added to the complement $A G$ , the other to the complement $G E$ .

Book II. Prop. XIII. This may be proved more simply thus:—

∵ (in fig. 1.) $B C$ is divided into 2 parts in D,

and ∵ (in fig. 2.) $B D$ is divided into 2 parts in C,

∴ (in both) $B C^{2} + B D^{2} = 2 B C . B D + C D^{2}$ ;

add to each $A D^{2}$ ;

∴ $B C^{2} + B D^{2} + A D^{2} = 2 B C . B D + C D^{2} + A D^{2}$ ;

but $B D^{2} + A D^{2} = A B^{2}$ ,

and $C D^{2} + A D^{2} = A C^{2}$ ;

∴ $B C^{2} + A B^{2} = 2 B C . B D + A C^{2}$ ;

∴ $A C^{2} < B C^{2} + A B^{2}$ by $2 B C . B D$ .

Also ∵ (in fig. 3.) $A B^{2} = B C^{2} + A C^{2}$ ,

add to each $B C^{2}$ ;

∴ $B C^{2} + A B^{2} = 2 B C^{2} + A C^{2}$ ;

∴ $A C^{2} < B C^{2} + A B^{2}$ by $2 B C^{2}$ , i. e. by $2 B C . B C$ .

Therefore in any triangle, &c. Q. E. D.

List of Abbreviations

$/ A B$	means	the line $A B$
$∠ B A C$	”	the angle $B A C$
∟	“	right angle
⊥	“	at right angles to
⊙	”	circle
▵	”	triangle
∥	”	parallel to
∦	”	not parallel to
=	”	equal to
≠	”	not equal to
>	”	greater than
<	”	less than
≯	”	not greater than
≮	”	not less than
□	”	parallelogram
$A B^{2}$	”	the square described on $A B$
$A B . A C$	”	the rectangle contained by $A B$ , $A C$
∵	”	because
∴	”	therefore
Q. E. F.	”	quod erat faciendum
Q. E. D.	”	quod erat demonstrandum

N.B. The symbol for “less than” must be carefully distinguished from that for “angle.”

The symbols for “greater than” and “less than” may be distinguished from each other by remembering that the greater end of the symbol is placed next the greater quantity. Thus “ $A > B$ ” shows that A is the greater, “ $A < B$ ” that B is the greater.

The symbols for “because” and “therefore” may be distinguished thus: when “because” is used, the pyramid (∵) is balanced on the point, to show that the argument is still unsettled; but when “therefore” is used, the pyramid (∴) is placed on the base, to show that the argument is settled.