Ex. 6. In a triangle two sides are 18 and 23 miles, and they include an angle of 58°24′36′′. What is the spherical excess? Answer 2" 31639.

Ex. 7. The length of a base measured at an elevation of 38 feet above the level of the sea is 34286 feet: required the length when reduced to that level? Ans. 34285.9379.

Ex. 8. Given the latitude of a place 48°51′N, the sun's declination 18°30′N, and the sun's apparent altitude at 10h11m268 AM, 52°35′; to find the angle that the vertical on which the sun is, makes with the meridian. Ans. 45°23′28.

Ex. 9. When the sun's longitude is 29°13′43′′, what is The obliquity of the ecliptic being 23°27′40′′.

his right ascension? Ans. 27°10'13".

Ex. 10. Required the longitude of the sun, when his right ascension and declination are 32°46′52′′, and 13°13′27′′N respectively. See the theorems in the scholium to prob. 12.

Ex. 11. The right ascension of the star a Ursæ majoris is 162°50′34′′, and the declination 62°50'N: what are the longitude and latitude? The obliquity of the ecliptic being as above.

Ex. 12. Given the measure of a degree on the meridian in N. lat. 49°3′, 60833 fathoms, and of another in N. lat. 12°32′, 60494 fathoms: to find the ratio of the earth's axes.

Ex. 13. Demonstrate that, if the earth's figure be that of an oblate spheroid, a degree of the earth's equator is the first of two mean proportionals between the last and first degrees of latitude.

Ex. 14. Demonstrate that the degrees of the terrestrial meridian, in receding from the equator towards the poles, are increased very nearly in the duplicate ratio of the sine of the latitude.

Ex. 15. If p be the measure of a degree of a great circle perpendicular to a meridian at a certain point, m that of the corresponding degree on the meridian itself, and d the length of a degree on an oblique arc, that are making an angle a with the meridian, then is d = tion of this theorem.

рт
p+ (m − p) sin.2 a'

Required a demonstra

THE GEOMETRY OF CO-ORDINATES.

If any figure whose genesis is known be drawn upon a plane, and through a given point in that plane lines be drawn in any given directions; and if parallel to these lines, lines be drawn from any point whatever of the figure, and terminating each in that to which the other is parallel: then there will be, in every case, a constant relation between these last-drawn lines, which can always be expressed by means of an equation *.

[For example, in the case of plane curves :-let the angle be a right one, and the given point be the vertex A of a parabola, the one line drawn through it coinciding with the axis AH of the parabola, and the other with its tangent (see

In like manner, if any surface be given by means of its genesis, and through a given point planes be drawn parallel to three given planes; and if from any point of the surface three lines be drawn parallel to the three planes, each of which is terminated by the two planes to which the other two are parallel: then the relation between these three lines will be constantly the same, and may, in every case, be expressed by means of an equation.

fig. p. 142) RS. Then P denoting the parameter, we have by cor. 1. prop. i. the relation between AF and FG, (where AF is equal to the line drawn from G a point in the curve parallel to AH) expressed by the equation P. AF = FG2. Or, in the circle, (see fig. Geom. theor. 87, vol. i.) we have the equation AD. DB = DC2, where AB is the diameter of the circle, and AD is equal to the line drawn from C parallel to AB, and terminated by the tangent drawn to the circle at A.

Or again, in the ellipse and hyperbola we have (theor. i. cor. i. of each) AD. DB: DE2:: AB2: ab2; or, as an equation,

DE2=

ab2 AB2

ab2
AD. DB; or DE2 = (CD2 CA2).
AB2

These examples, which are very simple, and with which the student is already familiar, will be sufficient to explain the fundamental idea of this branch of Geometry; and we proceed to lay down a few definitions and first principles.]

DEF. 1. The lines drawn through the given point (A in all these cases) are called the axes of co-ordinates; or the axes of reference, since to these all the parts of the figure are referred.

DEF. 2. The lines drawn from the points of the curve parallel to the axes of reference, are called the co-ordinates of those several points.

[The segments of the axes intercepted by these lines are also called the coordinates, since they are equal to them in magnitude.]

DEF. 3. The point in which the axes intersect (A) is called the origin of coordinates, since it is from that point that their several lengths are estimated.

DEF. 4. The angle under which those axes intersect is called the angle of coordinates. When that is a right angle, the co-ordinates are said to be rectangular, and when that angle is oblique, the system is said to be oblique.

DEF. 5. The equation which expresses the particular relation existing between the co-ordinates of any curve, is called the equation of that curve.

DEF. 6. The co-ordinates are usually expressed in the equation by x and y, and the given parts of the figure by other single letters as in the "Application of Algebra to Geometry," in the first volume.

[If AF, fig. prop. ii. Parabola, p. 143, be denoted by a, the equation of the parabola referred to AD as the axes of x, will be y2 = 4ax.

x) x.

If AB, fig. theor. 87, Geom. p. 326, vol. i. be denoted by 2a, then the equation of the circle referred to AB as the axes of x will be y2 =(2a In a similar manner in the ellipse and hyperbola (where AB = 2a, and ab = 26), the equations are respectively

If the origin had been assumed in a different place, the inclination of the axes of co-ordinates to each other and to the axes of the curves, the equations would have been of a different form, and in most cases more complex. The degree or dimension of the equation is, however, not altered by these circumstances.]

DEF. 7. All curves are called lines, so that the straight line may be included under one common denomination with them: and they are also called in reference to their equations, the loci of those equations.

DEF. 8. Curves are said to be of different orders according to the order or degree of the equation which expresses the relation between their co-ordinates.

Thus, if the equation be of the first degree (or ax + by + c = 0) its representative locus is called a line of the first order: if of the second degree, as ax2 + bxy + cy2 + ex + fy + g = 0, the locus is a line of the second order: and

so on.

Those coefficients may be + or -, or 0; still the locus is of the same degree, so long as the degree of the equation is not lowered by those particular values given to the coefficients.

[It will presently be shown that the straight line is the locus of an equation of the first degree: and it is obvious that each of the conic sections (vide p. 185), is a locus of some equation of the second degree. It will hereafter be shown that every equation of the second degree expresses some conic section.]

PRINCIPLES.

P

R

Ꭱ

Q

Ο

Q

Ps

R

P

1. Let X'X and Y'Y be the axes of reference, O, their point of intersection, the origin of co-ordinates. Then it is usual to consider the ordinates which are measured on the axis of a on the right hand as + and then (as was the case with the cosines in Trigonometry, vol. i. page 384.) those which are measured on the left as In a similar manner, the ordinates measured above O on OY are taken +, and consequently, those below will (like the sines in Trigonometry)

be

2. The position of a point is given when its distances from the two co-ordinate axes (measured on lines parallel to those axes) are given, with the signs prefixed to those distances which indicate on which sides of them it is situated. Thus if OQ = RP, = Չ = and OR = P,R=r, were the lengths of the co-ordinates: then the point P, is defined by the co-ordinates q, r; P2, by -q, r; P, by - q, —r ; and P1 by

q, γ.

119

[These quantities, for the sake of designating their being co-ordinates are generally designated by x and y; and to shew that they are given quantities, they are marked by an accent, (or if there be several sets, each set by a different number of accents) placed below them :-As x,y,, x,y,,, &c. Sometimes also instead of x,y,,,„„Y„„,, &c. or x′′y", "y"", they are simply denoted by x2Y2, XзYз, &c. Generally, too, where these quantities are supposed to involve their own signs, the comma, as between the q and r above, is entirely dropped: but cases occur in which this cannot be done without creating ambiguity. This is especially the case when the values are given in numbers, in which case the sign must be prefixed: and then if the commas were omitted, the signification would be dubious, if not false.

CHAPTER I.

THE STRAIGHT LINE.

I. To find the distance between two points whose co-ordinates x,y, and

given.

From the annexed figure, where PyP, = - a, the lines P,Q,, P,R,, and PQ, PR, being the co-ordinates of P, and P,, respectively, by Trig. vol. i. p. 404,

π

P, P., ± √x,

- x,,2)2+2(x,—x,,)(y, — y,,) cos. a + (y,—y,,)2 when a =

it becomes simply P,P„ =

±√(x, − x,)2 + (y, — y,,)2 . .

y=x tan. ẞ, and y = x tan. ẞ + c

(8)

The student should trace the mutations which arise from the variations of ß.

III. Given the equation of a straight line, and its angle of co-ordination, to construct it. (See last figure.)

Since the equation holds good for all values of x, it is true when x = 0; and in this case, taking (5) we have y = —

c'

:

Set off this value on the axis OY above or below as the sign + or – belongs to it. This will indicate one point in the line to be constructed.

In like manner it is true for all the values of x which correspond to values of

C

α

Set

y in the equation (5); and therefore when y=0. But in this case x = — off this on the axis of x, to the right or left as it is+ or -. Then this gives another point of the line.

The line drawn through these two points is the locus of equation (5.)

It is obvious, however, that if we had taken any two values of x or y, and found the two corresponding values of y or a respectively, and constructed the points, then the line through these would have been that sought. The advantage of the method first indicated is-that the calculation and the construction are both less laborious than in the latter one.

IV. To find the equation of a straight line passing through two points whose co-ordinates are given.

[Here the object is to discover the coefficients of the equation.]

Denote the given co-ordinates by xy, and x,y,, respectively.

"

Then, the general form of the equation is given in (7) in two ways. Take the latter and since xy, and x,y,, are known values of x and y in it, we have y, = ax, + b, and y„, = ax„, + b. From which we have

"

It is sometimes more commodiously written in the equivalent form

V. To find the equation of a line which passes through a given point and is parallel to a given line.

Then the equation of

inclination is given in the problem. Put it equal to k.

the line is, (x

VI. To find the co-ordinates of the point of intersection of two straight lines whose equations are given.

At this point, x and y have the same values. If we denote the equations generally by ax + by + c, = 0, and ax + by + c,, = 0, we have as usually found in simultaneous equations.