bases cx and XD would be equal), and that therefore the cistern has been emptied of much more than half its contents; but we are not informed by that Article, nor by any other of PART I., how much greater the former parallelogram is than the other, nor therefore how much more of the fluid has been let off than retained. This, however, is done by the present Article: As CD is ten feet and a half, and XD one foot and three-quarters, cx must be eight feet and three-quarters; and, therefore, cx must be five times as long as XD. Hence, as, by ART. 94, the parallelogram CIGX has to the parallelogram GXDE the same ratio as the base cx has to the base XD,-the former parallelogram must be five times the breadth of the latter, and, consequently, we know that the oblong body of water drawn off was five times the depth of that which remains, or, in other words, that five times as much fluid has escaped as is still in the cistern.

In the same manner, by stepping, or measuring in any less erroneous way, the length of a Canal, supposed equally wide and deep for the length stepped,-we may determine whether the quantity of water between two locks, or any two imaginary partitions, be equal to that between any

lines, while the former denoted such as are divided as above for the purposes of measurement.

The Common Scale, or Foot-Rule, is a flat-sided bar, of ivory or box in general, whose whole length is divided into twelve equal parts, called inches. The Carpenter's Scale, or Sliding Rule, consists of two pieces, each a foot in length, connected by a folding joint; with a third piece which slides in the face of one of the former; all three divided into parts of a certain length. There is also a kind of scale in frequent use among carpenters, masons, &c. which consists of four pieces connected by three folding joints, which allow the extreme pieces to be doubled on the other two, so as to render the instrument more portable. Several other kinds of Scale are in use,-all of which, exhibiting on their surface one or more lines divided into a certain number of equal parts, may be considered as practical illustrations of this problem. They are called Scales, from the Latin word scala, a ladder, the equally-distant steps of which their marks of division resemble. Gradus is also the Latin word for steps, and hence these instruments are said to be graduated; the process by

other two; and if not, how much one quantity of water exceeds the other. For illustration's sake, if the two former be 100 steps asunder, and the latter 250,-we know by ART. 94, first, that the two contained bodies of water are not equal, and secondly, that there is twice-and-a-half as much in the one as in the other.

D

B

E

A Land-Surveyor having measured the area of a parallel-sided field ABCD, and found it to contain twelve acres, set about computing the surface of that which adjoins, BCFE, and which is also parallel-sided. But observing that the ditches, or boundaries, BE and CF are only continuations of AB and DC, and that CF is two-thirds of DC, he performs his computation in a much simpler manner, by help of ART. 94, than he could do without it. For, as the parallelograms ABCD, BCFE, have the same ratio as their bases DC, CF, and as the latter of these is two-thirds of the other, so likewise must the parallelogram BCFE be twothirds of ABCD; that is, it must contain exactly eight

acres.

From these few examples the student will readily per

which they are graduated being called graduation.

E

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If in the opposite unprinted space, which is about an inch and a half square, I wished to delineate with great accuracy the face of a House, or building of any kind, whose breath was 50 feet, height 30 feet, width of door 4 feet, of windows 3 feet, length of same 5 feet, the door being 7 feet high from the threshold. Let us see how I might proceed to do this by help of a graduated rule?

A

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I take a series of 50 such equal divisions on the scale as will, together, at least not exceed an inch and half. I mark off this length suppose AB on the blank space assigned. At either extremity of AB raising the perpendicular BD, by PROB. VII, I mark off a series of 30 of these divisions, suppose BC on BD from

ceive to what a profitable account he may turn his knowledge of the present article. Almost every object he sees will serve to exercise the principle of it within him; and it would be a long catalogue indeed which would exhibit the various instances from its application to which the mind would derive pleasure, though the worldly interest might not derive any tangible advantage. If the great end of life be happiness, it makes very little difference whether the items of that be procured by the use of our mental or corporeal faculties. A play-going Mechanic does half-an-hour's overwork for the price of a gallery-ticket; but if he loved Nature's theatre as well as the King's, he might by the same expense of time, in applying this principle to some natural object, gain,-not indeed a shilling, but as much pleasure of a different kind as no number of shillings could purchase. Which is the happier, he who sees Harlequin jump through a hoop, or he who can tell the utmost breadth of Harlequin's shoulders from seeing only an inch-piece of the hoop's circumference *?-There is no ground of comparison: the one,

B.

This gives the height of the roof-level Ec. For the width of the door I mark off 4 of the same divisions, suppose cm, and for its height 7, suppose ca. For the width and length of each window I mark off 3 and 5 of the same divisions respectively, suppose de and do. In this manner the building will be accurately represented so far (and by a similar process throughout all its details), on the blank space required.

The above is very simple. But in the first place, we may not have a scale; and, in the second place, even though we have a scale, its divisions may be so long, that 50 of them would exceed an inch and half. For example, the least division on the common scale, or foot-rule, is one-tenth of an inch; so that a series of 50 of these would be five inches in length. In the third place, we may like to know the secret of rendering ourselves independent of such an instrument. In short, for these and many other reasons, it is necessary that the student should learn how to construct a scale for himself with facility and precision. This is to be accomplished by means of PROB. XV., as follows (the delineations being restricted as above to a blank space an inch and half square):

By completing the inch-piece, or given arch of the hoop, into a circle, as described in page 93; and then taking the length of the diameter: this will be the utmost possible breadth of Harlequin's shoulders.

perhaps, feels no pleasure in witnessing feats of activity, and the other would not give a fig to know how he might

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Draw a right line xy somewhat less than an inch and half long; from either extremity x draw another right line xz making an angle with xx. On xz take any convenient length xa, which repeat 10 times on xz. Then take ks equal to kx, the sum of those 10 repeated lengths, and take ST, tu, uv, equal to ks, until xv be equal to 5 times kx. Join vy, and draw right lines from the points u, T, s, k, i, h, g, f, e, d, c, b, a, parallel to vy. Hence, by PROB. XV., the line xy is divided into 5 equal parts XM, MN, NO, OP, PY; and also the part xм is divided into 10 equal parts.

30

Here then we have constructed a scale xy for ourselves, which we may use frequently with more convenience than a graduated rule, and, indeed, very often when we could not use the latter at all. Thus: considering each of the smaller divisions as a foot, each of the larger divisions will represent 10 feet, and the whole line 50 feet. Consequently, if we take on the given blank space, AB equal to XY, BC equal to xм, cm equal to 4 of the smaller divisions, ca equal to 7, de equal to 3, do equal to 5;-we shall have represented the building so far, with an accuracy proportioned to the accuracy of the scale so constructed*.

* In order that the scale XY may be constructed with great accuracy, we recommend the elementary length xa to be taken as great as possible: and also the line xz to be drawn as nearly perpendicular to XY as possible. For by these means the parallels will be kept farthest asunder, which are evidently crowded in our diagram, from the want of room to follow our own recommendation.

measure the shoulders of Atlas himself. But it is plain that both may derive equal gratification from these very different sources.

ART. 95 is only a sort of converse of the present theorem, and may be applied to the same purposes:

"ART. 95. Parallelograms which have equal bases, have to each other the same ratio as their altitudes."

:

Thus the windows of the drawing-room and upper story are each three feet in width, but the former are six feet long, while the latter are but four feet six inches: how much less glass is required to furnish the one than the

other?

Answer: One-fourth. Because, by ART. 95, the area of the greater has to the area of the lesser the same ratio as their altitudes,—that is, the same ratio as six feet to four feet six inches. But the latter altitude is one foot six inches, or one-fourth, less than the former; consequently, the area of the upper window is one-fourth less than that of the lower, and will therefore require one-fourth less of glass to furnish it.

Again: the two side-walls of a house are thirty feet from front to rear, but one, being the party-wall of an adjoining house, is eighty feet high, while the other is but sixty: how many bricks more will the former require than the latter?

Answer: One-third. Because, by ART. 95, the areas of the walls have the same ratio as their altitudes; that is, the same ratio as eighty feet has to sixty. But the former altitude is twenty feet, or one-third, more than the latter ; consequently, the area of the higher wall is one-third more than that of the lower, and will therefore require one-third more of bricks to raise it.

ART. 97. "Two equiangular parallelograms which have the sides about their equal angles reciprocally proportional, are equal."

Here is another criterion of equality between parallelogrammatic figures. If they be equiangular, and if the sides about their equal angles be reciprocally proportional,