Def. The points in which lines forming a triangle intersect are called the corners of the triangle; the parts of the lines between

the corners are called the sides of

the triangle; and the parts of the lines not between the corners are called the sides produced of the triangle.

Def. The angles formed by the sides of a triangle are called the interior angles or simply the angles -of the triangle; and the angles formed by the sides, and other sides

produced are called the exterior angles of the triangle. Def. A plane figure, all points of whose boundary are equally distant from a fixed point within it, is called a circle.

Def. The fixed point within a circle, from which all points of its boundary are equidistant, is called its centre.

Def. The boundary of a circle is called its circumference.

Def. The distance between the circumference of a circle and its centre (measured by the line joining any point in the circumference to the centre) is called

A

its radius.

Note-By the nature of its definition all radii of the same circle are equal.

Ax. If the centre C of one circle is on the circumference of another circle, and a point A on the circumference of the first is within the circumference of the second, the

circles will intersect in two points.

Post. Let it be granted that a circle may be described with its centre at any given point, and its circumference at a given distance from that point.

Ax. If two or more magnitudes are equal to the same magnitude they are equal to each other.

Ax. If equal magnitudes are added to other equal magnitudes (or to the same magnitude) the sums are equal.

Ax. If equal magnitudes are taken from other equal magnitudes (or from the same magnitude) greater than themselves, the remainders are equal.

Note-On the use of drawing instruments implied in the Postulates.

The Postulate on the description of a circle implies that a pair of compasses is to be used, whose points will maintain the same distance apart, as one of them is swept round the circumference, the other being fixed at the

centre.

A

Incidentally also it assumes that compasses may be used for a limited

B

transference of distances: for if C is the centre, and CA the distance at which the circle is to be described; then if B is another point on its circumference, the compasses, in passing round from A to B, transfers the distance CA to CB.

And again, when the points of the compasses are at B and C, if we keep B fixed, and sweep out another circle, with the point at C, then if D is any point on the circumference of the latter circle, BD, BC, and CA are all three equal to each other; for we have never changed the distance apart of the compasses' points. So that, without doing more than the Postulate demands, we have transferred the distance CA to BD.

Now compasses will preserve the distance of their points apart just as well when they are lifted, as when they are used to sweep out a circle. So that Euclid's refusal (implied in Prop. 2) to permit them to be used to transfer distances, is an arbitrary and unmeaning restriction: moreover it is a restriction never adhered to in practice. We say therefore that the use of compasses is postulated for describing circles, and for the transference of distances. Cf. 'Syllabus,' p. I.

The Postulates on the drawing of a straight line are usually taken to mean that the use of an ungraduated straight-edge is permitted. But clearly this is not drawing a straight line in the same sense in which compasses draw a circle. The analogous mode of drawing a circle would be to make a circular disc, like a coin, and use it to trace round. And as in this case there would at once arise the question-How are we to make the disc circular? so in the other there arises the question—How are we to make the edge straight?

Curiously enough, until the year 1864 no mechanical way of drawing a straight line, similar to the mechanical way in which compasses draw a circle, was known. But in that year such a way was discovered by M. Peaucellier, a French engineer officer. The instrument he devised is known as Peaucellier's Cell. The learner will find it easy and interesting to make one for himself. The following is an outline of its mode of construction—

Take four bars of one length, and two of another length-the two may be shorter or longer than the four; but when the two are taken shorter the instrument is more compact, and works more freely.

Suppose, in the figure, that AQ, BQ, AP, BP are the longer bars; and that AO, BO are the shorter.

Pierce holes in them at A, Q, B, P, O; and connect them through these holes by pivots, all freely moveable, except the one at O, which is to be of the nature of a nail or screw to go into a board on which the instrument is placed. Now by means of another bar QC, of any convenient length, pivoted to the

others at Q, and to the board at C, the end Q can be made to move on the circumference of a circle. If C be so placed that CQ is equal to CO, then, as Q is moved about, P will go accurately along a straight line.

By experiment it will be found that, when CQ and CO are equal, so that P draws a straight line, Q can only be made to go a part of the way round the circle; but that, when CQ and CO are unequal, P describes a circle, and by properly arranging the distances, Q can be made to go entirely round one circle, and P entirely round another.

The theory of the movement will be found on p. 360.

ABBREVIATIONS.

The following symbols and contractions will be used as shorthand equivalents for the ordinary words printed in immediate connection with them.

Note-The symbol = is used solely as an equivalent for the words' is equal to'; and means only that the sum total of all that is placed before the symbol is equal to the sum total of all that follows it. The symbol does not imply any equality of part to part. Nor is the symbol ever used as the equivalent of the adjective 'equal.' Thus we write 'make A equal to B'; or 'A and B are equal.' Similar remarks apply to the symbols > and <.

The 's' after the symbols As, ||s, &c., is dropped in the singular; and the symbols ||,, are used both for the corresponding noun and adjective.

A few other symbols are introduced on pp. 26, 53, 111, 177, 233, 243. The period, to indicate the elision of a part of a word, as shown above, is convenient for printing; but, in writing a contracted word, it is best always to mark the place where the elision is made by an acute accent, in the manner following

st' for straight
rect' for rectangle

diag's for diagonals

p't for point
rem'g for remaining
alt's for altitudes

When one series of magnitudes is said 'to be equal to,' or 'to coincide with,' or 'to correspond to,' a second series of magnitudes, 'respectively,' this last word indicates that such equality, or coincidence, or correspondence, is true between the magnitudes of the one series and those of the other each to each, in the order in which they are named.

For example: if X, Y, Z are stated to be 'respectively' equal to A, B, C, this means that X = A, Y = B, and Z = C.

N. B.-Until the learner is thoroughly familiar with the sequence of the main geometrical truths, as linked together by Euclid, he should write (in pencil) opposite any conclusion, depending on a previous Prop., the reference to that Prop.: e. g. on p. 12, line 14 depends on the 4th Prop. of Book i.; which should be indicated thus... ▲ DBCA ACB. (i. 4).

Proposition 1.

PROBLEM-On a given finite straight line to construct a triangle whose three sides shall be equal.

A

Let AB be the given st. line.

With A as centre, and AB as radius, describe a ; and with B as centre, and BA as radius, describe a O.

Suppose C one of the pts. in which the Os cut; and join CA, CB.

Since AC and AB are radii of the same O,

i. e. AC and BC are each equal to the same AB.

i.e. the three sides of the ▲ ABC are equal.

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