(10) Define a right angle, and exhibit it. Can one right angle be greater than another right angle? What is the way of determining whether one angle is greater than another?

(11) What are the names by which certain angles are distinguished? Exhibit an angle of each sort.

(12) Explain clearly the difference between the angle ABC and the triangle ABC.

(13) By what names are triangles distinguished according to their form? Exhibit a triangle of each sort.

(14) Does the magnitude of an angle depend upon the magnitude of the lines by which it is formed?

(15) How many lines are necessary to form an angle? How many to form a triangle?

(16) How many angles are there in a triangle? Does the magnitude of a triangle depend upon the magnitude of the lines which form its three angles?

(17) Does the word triangle mean 'three angles' in such a sense as to signify that the triangle is made up of the three angles, so as to be equal to them?

(18) Define parallel straight lines; and give an example.

If a straight line were drawn on the ceiling, and another on the floor, these two lines being produced ever so far both ways would never meet. Would they necessarily be parallel? Does the definition exclude such?

(19) How many kinds of parallelograms are there? What is the distinctive character of all, and of each? Exhibit each separately, and fully describe it.

(20) How many letters are used to denote a particular parallelogram, and where are they placed? Give an example.

(21) What is meant by the 'base' of a parallelogram? (22) Define a 'circle'; and explain clearly the difference between a circle and the circumference of a circle.

(23) How many letters are required to denote an arc of a circle? Why will not two serve, as in the case of a straight line? Where are the letters placed?

(24) What is the object of Euclid's three Postulates? (25) Upon what axiom does the Equality of geometrical magnitudes depend?

(26) Can one angle be equal to two other angles, or to three? Explain clearly.

(27) Is it possible for a triangle to be equal to a square? If so, say how.

(28) How many angles are there in a parallelogram? Is the parallelogram equal to the sum of its angles?

(29) Is a semi-circle a line or an area ?

(30) Is an angle an area? If so, how do you stand the statement at the end of (9) page 5?

under

(31) Can one triangle be added to another? If triangles be added together will the resulting sum necessarily be a triangle?

(32) Is a triangle equal to the sum of its three sides? (33) What is the difference between an angle, and a corner? What is the geometrical name for the latter?

EXPLANATION OF TECHNICAL TERMS USED
IN GEOMETRY.

(1) To 'describe' a certain geometrical figure, means to construct, or trace, it on a plane surface, as a board or sheet of paper.

(2) A 'given' line means a line 'given' sometimes in position, sometimes in magnitude, sometimes in both, according to circumstances; and the word 'given' means fixed or known.

(3) A 'proposition' is something proposed to be done; so that the heading of each separate article in the following section may be called a proposition. Sometimes 'propositions' are distinguished into two kinds; they are called problems, when something is proposed to be constructed or made; and they are called theorems, when some proposed statement is required to be proved.

(4) Corollary' signifies an after-conclusion beyond what is due, following obviously without any or much further proof from what has been already done or proved.

(5) à fortiori means by so much the more'. Thus, if A, B, C represent three geometrical magnitudes, and we know that A is greater than B, having proved that B is greater than C, we conclude, à fortiori, that A is greater than C.

(6) The 'converse' of a proposition is when the conclusion is turned into an assumption, and the previous assumption is made the conclusion. Thus to the proposition "The angles at the base of an isosceles triangle are equal to one another" the converse would be "Shew that, if the angles at the base of a triangle are equal to one another, the triangle is isosceles".

(7) 'reductio ad absurdum', (reducing to an absurdity), is a particular mode of demonstration often used by Euclid. It may be briefly explained by the following case-Required to shew that two geometrical magnitudes, represented by A and B, are equal to one another. We argue thus. If A is not equal to B, then A and B must be unequal. Suppose them unequal, and proceeding upon this assumption we arrive, by means of acknowledged axioms and legitimate reasoning, at an absurd conclusion, such as, for instance, that a portion of a magnitude is greater than the whole. If then the supposition that A and B are unequal legitimately leads to such a conclusion, it is plain that that supposition cannot stand; and therefore the only alternative is that A = B.

(8) To produce' a given straight line is to continue or extend it, so that the part added may be in one and the same straight line with the given line. Thus a radius of a circle, continued through the centre to meet the circumference again, until it becomes a diameter, is said to be produced.

(9) An'axiom' is a statement of an admitted truth, so plain and unquestionable as to need no demonstration, as long as words mean what they do; as that, for instance, "the whole of any magnitude is greater than a part of the same magnitude"-or, again, that 'two is greater than one'. Such truths do not specially belong to Geometry, but are practically interwoven with almost every operation of daily life.

STRAIGHT LINES AND RECTILINEAL PLANE

FIGURES.*

23. PROPOSITION I. To describe an equilateral triangle upon a given straight linet.

Let AB be the given straight line, which is to be one side of the triangle; with centre A and radius AB (POST. III. 20) trace a portion of the circumference of a circle on that side of AB on which the triangle is required; with the same radius and with centre B trace another portion of the circumference of a circle on the same side of AB, and intersecting the former in the point C; join the points A and C by the straight line AC (POST. 1.), and B and C by the straight line BC; then ABC shall be the equilateral triangle required.

A

For since B and C are points in the circumference of the same circle whose centre is A, AB= AC, (Def. 16); again, since A and C are points in the circumference of the same circle whose centre is B, AB or BA = BC;

.. AC=AB= BC,

or the three sides of the triangle ABC are equal to each other, that is, ABC is an equilateral triangle and it is described upon the straight line AB.

24. PROP. II. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles formed by those sides equal to one another, they shall also have their bases, or third sides, equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, viz. those to which the equal sides are opposite.

Let ABC, DEF be two triangles, in which the side

* A rectilineal plane figure means a plane surface (7) bounded by straight lines. According to the number of such lines, forming its boundary, each figure receives its distinctive name.

The Author does not deem it advisable to deviate much from Euclid's mode of expression, but rather to explain it, when it appears necessary, in a note. Thus, in this instance, to describe' a triangle means to construct or trace it; and 'upon a given straight line' means so as to have that straight line for its BASE. Also a given straight line' means here a line fixed both in position and magnitude.

triangle DEF, in such manner that the point A is upon the point D, and the line AB upon DE; then the point B will fall

B

upon E, because AB = DE._ Again, since AB falls upon DE, AC will also fall upon DF, because <BAC=¿EDF, (8.) Since, then, the point A is upon the point D, and the line AC upon DE, the point C shall fall upon F, because ACDF. Hence, since B is upon E, and C upon F, the line BC must coincide with EF, because BC and EF are straight lines between the same, or coincident, points. Therefore the triangles coincide, and are equal, in all respects, as stated above.

COR. Hence, also, if two triangles have the three sides of the one equal to the three sides of the other, each to each, in the same order, the two triangles will be equal, and their angles likewise will be equal, each to each, viz. those to which the equal sides are opposite. For it is evident from what has been shewn above, that such triangles, applied to each other as in the former case, will coincide in every part, and therefore be equal in all respects*.

25.

PROP. III. To bisect a given angle, that is, to divide it into two equal angles.

Let BAC be the given angle; it is required to bisect

it. In AB take any point D, and with centre A and radius AD describe an arc of a circle cutting AC in the point E; join the points D, and E, by the straight line DE, and upon

DE describe the equilate- A

B

D

E

EUCLID does not seem to have considered this sufficiently evident,

and therefore proves it by the process, usually called reductio ad ab

surdum, before explained.

† Given, that is, by being traced on a given plane surface.