Scholium. These spirals are of the same kind, as those formed by winding a chord around a conical spire, from the vertex to the base, in such manner, as to encircle the spire at equal distances; the exact length of such curve is of difficult determination.

The same spiral would be represented by the convolutions of a conical screw; also, by a screw represented on a disc. If the origin of the spiral is

at any point M, not in the centre of the concentric circles, then the area AFCM123A between the spiral and the outer circumference is of the product of the arc ACB through which the curve would have passed from the centre I, multiplied by the radius of the arc LMXIM (arc LM + arc BC) XMC.

C

D

If PQM be a triangle, whose base PQ = the semi-circumference ACB the angular space passed through by the two spirals MA, MB, then either portion PCM, QCM of the triangle may be expressed by PC or ACXCM = arc CBX IC arc LMXIM-(arc LM + arc BC) x MC, from this

substract the expression for the area included within the spiral, and the arc AB, and we have ACB × IC—¿LM × IM = the difference of the areas.

THE CYCLOID.

If a circle EPF be made to roll in a given plane upon a straight line BCD, the point in the circumference which was in contact with B at the commencement of the motion, will, in a revolution of the circle, describe a curve BPAD, which is called the cycloid.

This is the curve which a nail in the rim of a carriage-wheel describes during the motion of the carriage on a level road. The curve derives its name from two Greek words signifying "circle formed."

The line BD which the circle passes over in one revolution is called the base of the cycloid; if AQC be the position of the generating circle in the middle of its course, A is called the vertex and AC the axis of the curve. The description of the curve shows that the line BD is equal to the circumference of the circle, and that BC is equal to half that circumference. Hence also if EPF be the position of the generating circle, and P the generating point, then every point in the circular arc PF, having coincided with BF, we have the line BF = the are PF, and FC = the arc EP or CQ;

Draw PNQM parallel to the base BD.

Let A be the origin of the rectangular axes,

Ꭰ

AC the axis of x, and O the centre of the circle AQC.

Let AM = x, AO = α,

0:

MP y, angle AOQ = 8:

=

then by the similarity of the position of the two circles, we have

·. MP = PQ + QM = NM + QM = FC + QM = arc CQ + QM that is, y = a + a sin. ◊ = a (0 + sin. ◊)

x = a - a cos. = a vers.

(1)

(2)

The equation between y and x is found by eliminating be

tween (1) and (2)

But we can obtain an equation between x and y from (1) alone; that is from the equation, AP arc CQ + QM. For arc CQ a circular arc whose radius is a and versed

sine x

=a

=

{a circular arc whose radius is unity and vers. sin.

=

..y a vers.

If the origin is at B, BR and RP = y, the equations are x = a o --- a sin. 8

y

We shall not discuss these equations at length, as the me chanical description of the curve sufficiently indicates its form.

The cycloid, if not first imagined by Galileo, was first examined by him; and it is remarkable for having occupied the attention of the most eminent mathematicians of the seventeenth century.

Of the many properties of this curve the most curious are, that the whole area is three times that of the generating circle, that the arc CP is double of the chord of CQ, and that the tangent at P is parallel to the same chord. Also that if the figure be inverted, a body will fall from any point P on the curve to the lowest point C in the same time; and if a body falls from one point to another point, not in the same vertical line, its path of quickest descent is not the straight line joining the two points, but the arc of a cycloid, the concavity or hollow side being placed upwards.

BOOK V.

ON THE PRODUCTION AND RESOLUTION OF GEOMETRICAL
MAGNITUDES, CONSIDERED AS LINES, SURFACES, AND
SOLIDS, EXISTING IN THEIR SPECIFIC RELA
TIONS OF FORM AND PROPORTIONS.

CHAPTER 1.

DEFINITIONS AND PRINCIPLES.

ART. 1. We have hitherto referred lines, surfaces and solids, in all their varieties of figures and species, to some specific quantities and relations which were cognizable in such magnitudes, and whose properties were rendered evident to our consideration. Magnitude we have compared with magnitude; figure with figure; and we have thereby established their relations, under arbitrary considerations.

We will now consider magnitudes in the relation of their organization, or in the relation of their laws of production; and instead of referring magnitudes to specific magnitudes arbitrarily chosen, we will refer them to others, only in the rela tion of their laws of generation.

2. Since a point by definition is locality without extension, any number of associated points cannot possess magnitude, hence a magnitude is not a multiple of one, or any number of points.

3. Neither can any number of lines, however associated, constitute a surface, since lines are supposed to possess no breadth or thickness, one of which is essential to a surface; for if one line does not possess breadth, neither can any number of associated lines; and if a line be multiplied by any abstract number, since it is expressed only in relation to its length, it can only be multiplied or increased in that relation.

4. So, also, if a surface be multiplied by any number, in itself considered, the product cannot be a solid; for since the surface possesses no thickness, it does not possess the characteristic of the solid, and hence any number of such surfaces, or multiple of such surfaces in themselves considered, cannot be a solid.

5. The distance between any two points is a line. For a point being locality without extension, if there be two locali

ties, they must be seperate from each other, and their distance from each other is necessarily extension in space, which agrees with our definition of a line, viz., "extension in one dimension."

6. Space is a medium in which all positive objects, and all local relations exist; its existence is only indicated by its universal property of extension; it is infinitely divisible in each or all its three dimensions of extensions, and infinitely extensible.

7. Any definite portion of space, or any extension in space, is magnitude.

Magnitude may possess extension in one, two, or three dimensions, but space can properly exist only in its three dimensions of extension; if it can be divested of extension in one dimension, it can in another, and so on till its extension is extinguished. Magnitude may be properly applied to extension in whatever degree it exists; but space cannot properly exist independent of its three dimensions, which are its essential properties.

8. If there be two points A, B, the first point A, drawn through the distance AB, produces or describes

the line AB; that is, the distance from A to B AB in the portion of space passed through by the

point A, or the locality occupied by the point A in its passage, is the line AB.

9. If a line be moved through any space in a direction not agreeing with its length or extension, the locality passed through by the line is a surface.

CA a

CAC

Thus, if a line AB, be moved from its position AB to CD, it will, by that means, generate the surface ABDC, for the line will have occupied every portion of the extension between the two lines AB and CD, which cannot be said of any limited number of lines placed in juxtaposition across the figure ABDC. If, in this motion, the line AB always maintains its parallel position, and if any point A in the line, describes a right A line AC, the surface will be a rhomboid or parallelogram; and if, in addition to this, the line AC, or the direction of its motion is perpendicular to AB, then will the rhomboid be a rectangle. But if the line AB in its motion should not preserve its par allel position, or if the distance BD, passed through by the point B, is greater than AC, then the figure generated will depend on the nature of the lines which serve as its boundaries, but in general, in such case, one or both of the lines AC, BD will be

mohon Bb

D B