the magnitude and position of AB, with respect to AB: but

Sum and difference

then ar would denote AB,, a'r would denote AB, and a'r would denote AB, thus passing twice round the circle before we reach AB.

If we produce BA to D, and join B, B2, which bisects AD in of two lines E, we shall find

when con

sidered

with refer

2

ference to AB or r, it will be represented both in magnitude

√3

and position by r√1 (Art. 733), and EB, by - √3r√=1:

2

or in other words, that the symbolical sum of the two sides
AE and EB, of the right-angled triangle AEB, is the hypo-
thenuse AB1, the three sides being considered in magnitude and
also in position with respect to the primitive line AB: and in
a similar manner, the symbolical sum of AE and EB,
when con-
sidered both in magnitude and position with respect to AB, is the

For since the angle B1AD is 60o orrd of a right angle, the triangle B1AD is equilateral, and the perpendicular B, E bisects the base: it follows likewise that AB = AE2 + EB12, and therefore EB, AB, — AE2 = AB2 — † AB2 = { A B2,

=

hypothenuse AB, of the right-angled triangle AEB,, where EB, is drawn in an opposite direction to EB1, and therefore affected with an opposite sign*: this is a case of a general proposition, which will be established in a subsequent Chapter (xxx1), by which it will appear that if a and b√-1 represent two sides AB and BC (where the position of BC is referred to AB) of a right-angled triangle ABC, their symbolical sum or a + b√-1 will be the hypothenuse AC: and simi- A larly if a and b√-1 or AB and BC' be the two sides of a right-angled triangle ABC', their symbolical sum or a-b√-1

(or the symbolical difference of AB and BC), will be AC'. Again, it will be found that

B

and

or in other words, the symbolical sum of the two sides AB, and AB, of the rhombus AB, DB, is the diagonal AD which they include, and their symbolical difference AB1 – AB, is the second diagonal B1 B, which is at right angles to the former: the same proposition will be shewn hereafter to be true with respect to the sides and diagonals of any rhombus whatsoever.

It will be found however to be impossible to demonstrate generally these and other important propositions, or to make any extended applications of Symbolical Algebra to Geometry, without the aid of a knowledge of the theory of angles and their measures, and of the various periodical ratios which constitute the science of Trigonometry, or more properly Goniometry, and which we shall proceed to consider at length in the Chapters which immediately follow.

The lines EB, and EB2, being drawn in opposite directions, are affected with opposite signs, one + and the other - the application of these signs is independent of the sign √√-1, which farther indicates that EB, is perpendicular to AB with reference to which it is estimated.

CHAPTER XXVI.

The

Goniome

ON THE REPRESENTATION AND MEASURES OF ANGLES.

735. THE theory of angles, their measures, and the perisciences of odical ratios which determine them, constitutes the science of try, Trigo- Goniometry, whilst the specific application of some of its results and Poly- to the determination of the sides, angles, and areas of triangles gonometry. would be properly termed Trigonometry, and to rectilineal

nometry,

Angles in

considered

ence to

their mag

and not

figures in general Polygonometry: but it has arisen from the associations connected with the progress of our knowledge of these sciences, that the least general of these denominations has anticipated and superseded the adoption of the others, and it is usual to include, under the name of Trigonometry, the science of Goniometry in its most extended applications.

736. Angles are considered in Geometry, as absolute magGeometry nitudes only, without any reference to their mode of generation: with refer- they may possess every magnitude between zero and two right angles, which are their limiting values: for lines which contain nitude only, no geometrical angle with each other, and which, in conformity with other views of the generation of angles, make angles with their mode each other equal to zero, or two right angles, or any multiple of genera of 2 right angles, are either parallel to each other, or in the same straight line: and such lines are not distinguished from each other in Geometry, as being drawn in the same or in opposite directions*.

with respect to

tion.

Angles may be con

ceived to be

737. As lines, however, may be conceived to be generated by the motion of a point (Art. 558), so likewise angles may generated be conceived to be generated by the motion or revolution of by the revolution of

a line

round a

In applying the principles of Symbolical Algebra to the representation of lines, we have been enabled (Arts. 561 and 562) to form two classes of parallel fixed point lines (including those which are in the same straight line), according as the lines and from a given posi- which form them are drawn in the same or in opposite directions, or according as tion. they form angles with each other, which are an even or an odd multiple of two right angles, zero being included amongst the former: this classification of parallel lines is not recognized in Geometry, where angles cease to exist at the extreme limits of zero or two right angles.

a line round a fixed point: when viewed with reference to such a mode of generation, and not with respect to their magnitudes merely, angles will be found to be not only capable of indefinite increase, but likewise of affections which may be symbolized by the ordinary signs of algebra.

Thus, if a radius AP revolve from the primitive position

AB to the position AP, it may be said to pass over or generate the angle BAP, whilst the point P passes over or generates the arc BP: if the movement be continued from AP to AQ, the arc BP will be increased by PQ, and the angle BAP by QAP: and it will follow from a well-known proposition, that the angles. BAP and BAQ will bear to

each other the same propor

tion with the arcs BP and BQ, by which they are subtended in the circumference of the same circle.

and

738. If the subtending arc be the quadrant BC, the cor- Degrees responding angle BAC is a right angle: and if we suppose the minutes. quadrant BC to be divided into 90 equal parts, the right angle will be divided by the radii which pass through these points into 90 equal angles, each of which is called a degree and

scale..

if the arc subtending a degree be divided into 60 equal parts, The sexaeach of them will correspond to, or subtend, an angle which gesimal is th part of a degree, which is called a minute: if the arc subtending a minute, be divided into 60 them will correspond to an angle, which is and is called a second: and we may proceed similarly to other inferior units in the sexagesimal scalet, as far as we choose to extend them.

Euclid, Book vi. Prop. 26.

60

equal parts, each of
th part of a minute,

The sexagesimal division of the circle has prevailed since the time of Ptolemy or the astronomers who preceded him, (see the article "Arithmetic" in the Encyclopædia Metropolitana, p. 401), and is so intimately associated with our habits of thinking and speaking on such subjects, as well as with our astronomical VOL. II. instruments,

T

increase.

1

Angles thus 739. It thus appears that the angle generated will increase generated in the same proportion with the corresponding are which is capable of indefinite described or passed over, by the extremity P of the revolving radius AP: and consequently if one of them admit of indefinite increase, so likewise will the other: thus, if P advance to C (Fig. Art. 737), the extremity of the quadrant BC, the angle generated is the right angle BAC or 90°: if P move onwards to P, in the second quadrant of the circle, the corresponding angle BAP, is greater than one right angle and less than two, bearing to the right angle BAC the same proportion that the arc BCP, bears to the quadrantal arc BC: the revolving radius AP will reach the position Ab, after describing two right angles BAC and CAb, and it is in this sense that the lines AB and Ab which are drawn in opposite directions, are said to make with each other an angle equal to two right angles*: if the motion of AP be continued, it will reach the position AP, after describing two right angles together with the angle bAP, or an angle equal to 180° + b AP2t: when it reaches c, at the extremity of the third quadrant, the generating radius has described or passed over three right angles, or 270°: if the movement be further

instruments, tables, and records, as to make its abandonment somewhat inconAdvantages venient and embarrassing: but the superior brevity and uniformity of processes of of the cen- computation adapted to the decimal scale is tending rapidly to replace the sexatesimal gesimal by the decimal, or rather by the centesimal, division of the degree: thus division of 130.27′. 42" in the sexagesimal scale is equivalent to 13o.4633 nearly in the the degree. decimal or to 13 degrees 46 (centesimal) minutes and 33 (centesimal) seconds nearly.

The French

drant.

Causes of

The French, simultaneously with the establishment of their Système métrique division of décimale, proposed to divide the quadrant into 100 degrees, the centesimal degree the quainto 100 minutes, and the centesimal minute into 100 seconds, and so on, and this division was adopted in the Mécanique Céleste of Laplace and other cotemporary scientific works. The change however from the nonagesimal to the centesimal degree, was attended with no advantages sufficient to compensate for the great sacrifices of tables and records which its adoption rendered necessary, and its use was speedily abandoned, even in France. If the proposed change had been limited to the centesimal division of the nonagesimal degree, it could hardly have failed, when the authority of the great men who proposed it is considered, to have been readily and universally adopted.

its failure.

* At this point the geometrical angle contained by AB and the revolving radius ceases to exist, inasmuch as the lines which contain it are in the same straight line no regard is paid in Geometry to the directions of lines when considered

per se.

The corresponding geometrical angle, which is formed by AB and AP, is BAP, or 360o — ( 180o + b AP ̧) = 180o — bAP,: this is sometimes called the supple ment of BAP to 360o.