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Put

the square root of the greater number, And y the square root of the less number; or

Put x+y= the square root of the greater, &c.

14. It is required to divide the number 18 into two such parts, that the squares of those parts may be to each other as 25 to 16. Let the greater part. Then 18-x- the less. By the condition proposed, a2: (18—x)2 :: 25: 16. Therefore, 16x2 25(18-a)

By evolution, 4x=5(18-x)

If we take the plus sign, as we must do by the strict enunciation of the problem, we find x=10. Then 18-x-8. ·

And (10): (8)2:25:16

If we take the minus sign, we shall find x=90.

Then 18-x=18-90=-72.

And (90): (-72): 25: 16; a true proportion, corresponding to the enunciation; but 18 in this case is not the number divided, it is the difference between two numbers whose squares are in proportion of 25 to 16.

15. It is required to divide the number a into two such parts that the squares of those parts may be in proportion of b

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Prob. 14, is a particular case of this general problem, in which a=18,b=25, and c=16; and substituting these values in the result, we find x=10, and x=90, as before.

If we take b=c, the two divisions will be equal, each equal to ja, when the plus sign is used; but when the minus sign is

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nator is contained in the numerator an infinite number of times.

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It may appear absurd, that the two parts, both infinite and having a ratio of equality, (which they must have, if bc) can still have a difference of a. But this apparent absurdity will vanish, when we consider that the two parts being infinite in comparison to our standards of measure, can have a difference of any finite quantity which may be great, compared with small standards of measure, but becomes nothing in comparison with infinite quantities. See (Art. 60.)

Application of the foregoing Problem.

(Art. 94.) It is a well established principle in physics that light and gravity emanating from any body, diminish in intensily as the square of the distance increases.

Two bodies at a distance from each other, and attracting at a given point, their intensities of attraction will be to each other as the masses of the bodies directly and the squares of their distances inversely. Two lights, at a distance from each other, illuminating at a given point, will illuminate in proportion to the magnitudes of the lights directly, and the squares of their distances inversely.

These principles being admitted

16. Whereabouts on the line between the earth and the moon will these two bodies attract equally, admitting the mass of the earth to be 75 times that of the moon, and their distance asunder 30 diameters of the earth?

Represent the mass of the moon by c,

and the mass of the earth by b,

their distance asunder by a.

The distance of the required point from the earth's centre, represent by x. Then the remaining distance will be (a—x).

Now by the principle above cited, x2: (a-x): b: c.

This proportion is the same as appears in the preceding gen eral problem; except that we have here actually made the application, and must give the definite values to a, b and c.

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=26.9, nearly. Hence, a-x=3.1, nearly.

If we take the second values for the two distances, from the

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These values show that in a line beyond the moon, at a distance of 3.9 the diameters of the earth, a body would be attracted as much by the earth as by the moon, and the value of (a-x) being minus, shows that the distance is now counted the other way from the moon, not as in the first case towards the earth; and the real distance, 30, corresponding to a in the general problem, is now a difference.

We may make very many inquiries concerning the intensity of attraction on this line, on the same general principle.

For example, we may inquire, whereabouts, on the line between the earth and moon will the attraction of the earth be 16 times the attraction of the moon?

Let x the distance from the earth.

Then a-x- the distance from the moon.

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с

The attraction of the moon at the same point will be

(a—2)2

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25.7 diameters of the earth beyond the moon.

Observe that the 4 which stands as a factor to c is the square root of 16, the number of times the intensity of the earth's attraction was to exceed that of the moon.

If we propose any other number in the place of 16, its square root will appear as a factor to c; we may therefore inquire at what distance the intensity of the earth's attraction will be n times that of the moon, and the answer will be from the earth in a line through the moon,

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The same application that we have made of this general problem to the two bodies, the earth and the moon, may be made to any two bodies in the solar system; and the same application we have made to attraction may be made to light, whenever we can decide the relative intensity of any two lights at any assumed unity of distance.

(Art. 95.) This problem may be varied in its application to meet cases where the distances are given, and the comparative intensities of light or attraction are required.

For example, the planet Mars and the moon both transmit the sun's light to the earth by reflection, and we now inquire

the relative intensities of their lights at given distances, and in given positions.

If the surface of Mars and that of the moon were equal, they would receive the same light from the sun at equal distance from that luminary; but at different distances equal surfaces would receive light reciprocally proportional to the squares of their distances.

The surfaces of globular bodies are in proportion to the squares of their diameters. Now let Л represent the diameter of Mars and m the diameter of the moon. Also, let R represent the distance of Mars from the sun, and r the distance of the moon from

the sun.

Then the quantity of light received by Mars may be expressed M2

by R2

by

m2

; and the relative quantity received by the moon

But these lights, when reflected to the earth, must be 7.2 diminished by the squares of the distances of these two bodies from the earth. Now if we put D to represent the distance of Mars from the earth, and d the distance of the moon, we shall M2 R2D2

have

for the relative illumination by Mars when the whole

enlightened face of that planet is towards the earth, and

the light of the full moon.

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When the whole illuminated side of Mars is turned towards the earth, which is the case under consideration, (if we take the whole diameter of the body,) it is then in opposition to the sun, and gives us light, we know not how much, as we have no standard of measure for it; but we can make a comparative measure of one by the other, and therefore the light of Mars in this position may be taken as unity, and in comparison with this let us call the light of the full moon æ.

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Therefore x=

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