more generally than we have as yet proceeded to show. It may be used to obtain copies of all sorts of figures, whether rectilineal or not, whether geometrical or otherwise. There will not, indeed, be the same accuracy of resemblance, but there will be quite enough for practical purposes. For example, if we wish to take off a small transcript of a Map, no more is necessary than to fasten our paper on that of the map by four pins at the corners; and drawing from a certain fixed point, like the c in our plan of fortification, straight lines towards the vertices of the several headlands, or promontories, creeks, bays, and other remarkable points of the original, to trace between these lines, on our paper, other lines as parallel as may be to the corresponding portions of the outline of the map. According as the lines through the fixed point are numerous, and the lines between them are drawn nearly parallel to the corresponding portions aforesaid, the transcript will more closely resemble the original.

Again: let us suppose that abcdefg were the head on an ancient Coin, of which we desired a copy in larger dimensions. We have only to choose a convenient point x behind the original, and drawing the tangents xa, xg, to pro

duce them so as that the copy may be of the size required. Through the remarkable points b, c, d, e, f, we also draw other straight lines from the same point x; and finally trace the forehead AB, the nose BC, the under part of the face CD, the chin and throat DE, the breast EF, and the base FG, by lines nearly parallel to the several portions ab, bc, cd, de, ef, fg, of the outline abcdefg. If the lines from x be sufficiently numerous, and the lines between them be traced with sufficient precision, the outline ABCDEFGH will closely resemble that of the coin proposed. And in the same manner the eye, mouth, ear, &c. may be copied in larger dimensions.

It is plain that in making such transcripts of figures not accurately rectilinear, we have no right to expect geo

metrical exactness; by the method described, we can only obtain a rough copy as far as it is executed on the scientifical principle, and its perfect resemblance must be owing to the skilfulness of hand with which the operator can finish his work. As far as the scientifical principle extends, it manifestly supposes that the outline of the original is made up of very small rectilineal portions contained between the lines from the fixed point, and that the corresponding portions of the outline of the copy are also rectilineal. But as these portions are not accurately rectilineal in the original, neither are they so in the copy, and therefore no such geometrical parallelism, as we have supposed, really exists between them. But the geometrical principle being used as far as possible, will leave very little to be accomplished by mechanical operation.

A little farther on will be shown the method not only of reducing or augmenting a figure to one that shall be similar, -but reducing or augmenting it in any given proportion. See ART. 118.

ART. 114. "If two sides of any triangle have respectively to two sides of another the same ratio, and likewise the angles contained by each pair of sides equal,-the other angles of the triangles will be also respectively equal."

A simple Corollary, or Deduction, from this theorem has been applied, in the most signally ingenious manner, by the most profoundly skilful Mechanic, to the most practically useful invention, of this or any other age,—the Steam Engine.-Let us only establish this fact, and we shall, perhaps, be spared all further necessity of declaiming upon the practical utility of Geometry.

To begin with the deduction:

If from any two points (B and c) of a given right line (AC), two parallel right lines (BD and CE) be drawn, which have to each other the same ratio as the corresponding distances

(AB and AC) from the extremity (A) of the given line,then this extremity will be in a right line with the two extremities (D and E) of the parallel lines.

DEMONSTRATION. Draw the right line AD, and also the right line AE. Now, as in the triangles ABD, ACE, it is granted that the side AB has to the side AC the same ratio as the side BD has to the side CE,-and also, that the angle ABD is equal to the angle ACE (ART. 14),—hence, by ART. 114, the angles BAD and CAE must be likewise equal. Consequently, the line AD must coincide with, or fall on, the line AE, and therefore the three points A, D, and E, must be in one and the same right line. Which was the assertion made above.

Now let us see the very remarkable use which was made of this principle in the construction of WATT's Doubleacting Steam-Engine. By an extraordinary coincidence it happens that, not only in the same machine, but in the identically same limb of the machine, two other geometrical principles are brought into action; and, as if this were not sufficiently apropos, it will appear that these three principles are taken one from each PART of our GEOMETRY; so that in describing a single function of the aforesaid omnipotent Engine, we have an opportunity of exemplifying the practical tendency of our Science in PARTS I., II., and III., successively.

E

H

Round a fixt centre c in the pillar xv, the arm CA plays in a circular arch. At B, the middle point of CA, is a pivot, and at A is another; round these pivots two equal rods, BE, AG, play freely. At the other extremities, E and G, of these rods are pivots, round which the extremities of a third rod, EG, equal to BA, play freely. -Now, here are principles of PART I. brought into action; for in the four-sided figure ABEG, each opposite pair

H

of sides, BE and AG, GE and AB, being equal, we can show by ARTS. 15 and 19 that the figure is a parallelogram,which is essential to the due operation of the engine *.

Again: Round the fixed centre D, in the beam DX, the rod DE, equal to BC, plays freely; as also does its other extremity round the pivot E.-Now it can be shown that, by the motion of the two rods CB, DE, round fixed centres c, D, although the extremities B, E, of the connecting rod BE, move in circular arches, the middle point F moves up and down in a straight line. And in this the properties of the Circle, or the principles of PART II. are manifestly involved t.

Finally the arm ca playing up and down, the rod DE

:

Thus: Suppose the diagonal BG drawn, which would form two triangles ABG, BGE, with two sides equal to two, and one common. Hence, by ART. 6, the corresponding angles, ABG, BGE, are equal; but these being alternate, the lines AB and GE are parallel, by ART. 15; and finally, adjacent extremities of equal and parallel lines AB, GE, being joined by the right lines AG, BE, these are likewise parallel, by ART. 19. Consequently, by DEF. XI., the figure ABEG is a parallelogram.

With all these geometrical principles involved,-the Steam-Engine alone, nay, a single limb of it,-is enough to confute all scepticism about the practical usefulness of Geometry!

† Of the proposition advanced in the text, we subjoin a practical demonstration for the reader's satisfaction:

a

Let CB, DE, be the rods playing round the fixed centres C and D, while the rod BE connects them. Let CB', DE', B'E', be another position of the three rods; and CB", de", b′′e", third. Divide BE, B'E', B"E", equally at the points F, F', F These middle-points will be all found to lie in the same straight line F"FF'.-In like manner, the middle point of the con

necting rod, in any other position of the connected rods, will be found to lie in the straight line F"FF'; which proves that the middle point of the connecting rod moves as was asserted.

It is to be understood, however, that this line F"FF' is not an accurate "right" line, mathematically speaking, though its deviation from perfect straightness is imperceptible in practice, when the rods are in due proportion, and their play round the centres so small, as it is in the SteamEngine.

Following that motion, the middle point of the rod BE moving up and down in a straight line, and the rod AG being always parallel to BE,-it follows from the deduction above stated, that the extremity G of the rod AG will move up and down in a straight line.

For, as CB is half of CA, and BF half of BE, or of AG, which is equal to BE, therefore BF has to AG the same ratio as CB has to CA; and moreover, BF is parallel to AG. Hence, by the said deduction, the parallels BF, AG, having to each other the same ratio as the corresponding distances BC, AC, from the extremity c of the line AC,-this extremity c will be in a right line with the extremities F and G. Now, as the pivot G is always in the same straight line with CF, and as CG has to CF the same ratio as CA to CB, i. e. as CG is always twice the length of CF, the pivot G will move in a line exactly parallel to that in which F moves; hence the pivot. G will move up and down in a straight line *, as was asserted.

We now perceive how many geometrical principles are involved in explaining and demonstrating that celebrated contrivance of the Steam-Engine, by which the pivot G

This is obvious enough at first sight, but it may be also thus demonstrated:

Suppose ff the straight line through which the middle-point F of the rod BE moves, that point lengthening or shortening its distances CF, co, cf, from the centre

G

c, so as to describe that straight line. Join cf by a right line, and produce cf until cg be equal to twice cf; then g will be the place of the point G when the middle-point F is at f. Join Gg by a right line, and in the triangles cfr, cgG, as the side cf has the same ratio to the side cg as the side CF has to the side CG, while the contained angle at c is common to the two triangles,-therefore, by ART. 114, the angle frc is equal to the angle gGC, and consequently, by ART. 16, Gg is parallel to Ff. Now, let o be the place of the middle-point F any where between Fand f; also produce co till it meets Gg in x. Then, as the triangles CFO, CG, are equiangular, cx has the same ratio to co as CG has to CF, by ART. 110; that is, cx is twice co, and therefore, when the middlepoint F is at o, the place of the pivot a will be x, a point in the parallel Gg. In like manner it can be shown that the pivot & will always be found in the right line Gg parallel to Ff; which proves that the pivot is carried up and down in a right line, as was asserted.