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long. Multiplying these numbers, we get 800, the half of which, 400, is the number of square-yards in the field.

Again: let there be a piece of

C

ground, of any rectilineal shape

whatever. The area of this may be found on the same principle. For: we have only to sup

B

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pose the figure divided into tri

E

angles ABC, CAD, DAE, to find the area of each as above, and the sum of these areas will be the whole area of the figure.

As for example: let the bases AC and AD be 50 and 70 yards respectively; also, let the perpendiculars BF, CG, EH, on them, from the vertices of the opposite angles, be 5, 15, and 20 yards respectively.

Then the area of the triangle ABC is half 5 times 50, or 125 square-yards.

And the area of the triangle CAD is half 15 times 70, or 525 square-yards.

And the area of the triangle DAE is half 20 times 70, or 700 square-yards.

Hence, the sum of these areas, or the area of the whole figure, is 1350 square-yards.

On the other hand, by means of this principle we might divide a piece of ground, or plane surface of any rectilineal form, into a given number of equal parts. For we have only to measure the whole surface as above; then divide the quantity so found into that number of equal parts, and cut off portions successively equal to one of them. For example, suppose it were required to divide a four-sided field ABCD of any shape into two equal parts.

Draw AC, and measure it, as also the perpendiculars BE, DF, upon it from the vertices of the opposite angles. Let these three

30

lines be respectively 50, 10, and 20 yards long.

B

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C

Then the area of the triangle ABC is half 10 times 50, or 250 square-yards.

And the area of the triangle ADC is half 20 times 50, or 500 square-yards.

Consequently, the area of the whole field is 750 square

yards.

Now, as the field is to be divided into halves, each must contain 375 square-yards. Wherefore, as ABC contains only 250 square-yards, we must cut off from ADC a portion containing 125 square-yards. But as ac is 50 yards long, if we cut off a triangle Arc from ADC, such that its altitude za may be 5 yards, then its area will be half 5 times 50, or 125 square-yards. Hence, the line ar divides the field into two equal parts.

Again: if it were required to divide the same field into five equal parts. As the whole area is 750 square-yards, each of the five equal parts must contain 150 square-yards. Hence, we have only to measure one of the sides, as AB, which suppose 30 yards long, and cutting off a triangle AUB such, that its altitude vw may be 10 yards, then its area will be half 10 times $30, or 150 square yards; so that the triangle ATB is one of the five equal parts required. Now, as the area of ABC is 250 square-yards, therefore the area of Avc is only 100 square-yards: so that in order to cut off another equal portion of the field, we must add to Avc a part of ACD, whose area may be equal to 50 squareyards. This is done as above, namely, by cutting off a triangle ayc such, that its altitude may be 2 yards. For then its area will be half twice 50, or 50 square-yards, as required. Consequently Avcy will be another fifth portion of ABCD, Finally, as AyD contains the three remaining fifth portions,

if

30

20:

50

we divide yD into three equal parts ys, sr, rd, and draw AS, Ar, the triangles Ays, Asr, Ard, will be all equal, by ART. 29, as they have equal bases, and the same altitude: consequently, they will be the three remaining equal portions of the field.

In the latter part of the foregoing process, it was easier to divide AyD into three equal portions by dividing yD into three equal parts, than to measure the length of Ay, and then cut off a triangle asq, with its altitude of such a length as to make its area equal to 150 square yards. But

this method might have been taken: the other was only preferred for its brevity. It is to be observed, likewise, that the field might have been divided as required by other lines drawn from other points in other directions, and cutting off other figures. The division is, however, or at least may be, always effected by the principle set forth in ART. 89.

LEARNER. I now see by what means the field represented in page 4 might have been divided equally amongst the four brothers.

TEACHER. Yes; and likewise how the equality of the portions A, B, C, and D, might have been recognised, when the figure had been accidentally divided as required. For it were only necessary to measure the separate areas by means of ART. 89.

The other Articles of this Lesson do not readily afford practical examples. Their use is intermediate rather than direct, being chiefly apparent in the demonstration of other principles, which principles are applied to practice. Indeed, the Doctrine of Rectangles, though at first view abstract and speculative, is virtually so far otherwise, that to it we are indebted for numberless results, the most beneficial in practice. It pervades almost the whole world of Science, and is the great instrument by which geometrical calculations are performed in Astronomy, Optics, &c. NEWTON'S PRINCIPIA, wherein the nature of Gravitation and the System of the Universe are developed, cannot be understood without it; and although scientifical problems are now, in many cases, resolved by a different method of computation, this still remains in extensive use, as may be seen by opening any work of Art or Science which is treated geometrically. So perpetually in truth does it occur, of such frequent utility is it in works of this kind, that it has afforded a subject for pleasantry and satirical remark to the poet:

Alas! that partial Science should approve
The sly Rectangle's too licentious love!

Loves of the Triangles'

* A poem by the late MR. CANNING.

In which the author alludes to the ubiquity of this figure throughout the world of scientific literature. It is not often that either Poetry or Mathematics is obliged to the other for assistance in spreading its fame; nor indeed are the authors in these several departments of literature very forward to render a mutual testimony of merits; but here truth appears to have been too strong for prejudice, and the mathematicians owe to poets a handsome compliment, which we have no doubt they will have the liberality, when they have the opportunity, to repay.

THE

GEOMETRICAL COMPANION.

PART III.

DEFINITION XXXV. "Two quantities are said to have the same ratio to each other as two other quantities have, when every submultiple of the first quantity is contained in the second, the same integral number of times that an equi-submultiple of the third quantity is contained in the fourth."

The above definition is that upon which the whole Doctrine of Proportion is founded,—a doctrine no less necessary for advancement in the Sciences, than those of the Triangle and Circle, Parallels and Rectangles, as exhibited in PARTS I. and II., were necessary for initiation. In the Arts, likewise, though we can proceed to a very great extent by a knowledge of the very simple doctrines last mentioned, yet, to reach perfection, we must thoroughly understand the doctrine of proportion. To convince himself of the practical utility of PART III., he who has read the abstract principles of it in our GEOMETRY, has only to observe how they are applied in the following pages: but even he who has not read those abstract principles, nor any other, may be convinced of the same thing if he has ever known what it is to have done a sum in the Rule of Three or Fractions. Both these Rules are derived from the preceding definition, and we believe it would be hard to advance a stronger fact than this, to establish the practical utility of any science whatever. It may be said, perhaps, that although the utility of the rules themselves is un

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