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definite altitude, as the length of the perpendicular may be different for each of the two triangles of which the figure is composed.
Of the various plane surfaces bounded by curves of regular outline,* the only one which can be adequately treated of in elementary geometry, is the circle. But as the modes of estimating the space included by two other curves, the ellipse and the parabola, require to be explained in an introduction to the principles of gauging, such account wiil be given of the leading properties of those figures, as may be intelligible to persons who possess no knowledge of the higher mathematics.
It is evident that the boundaries of plane surfaces may present an almost infinite diversity of forms, some consisting wholly of a succession of straight lines joined at certain angles (regular or irregular polygons) ; some, of lines of continuous curvature, no part of which lies in a straight direction ; and some of a mixture of straight lines and curves. Straight-lined figures, whatever be their shape, may all be measured exactly by resolving them into a number of triangles or quadrilaterals, and computing the area of each of these separately, but in hardly any case is it possible to determine with absolute precision the magnitude of a figuro which is bounded by curved lines, regular or irregular, or partly by curves and partly by straight lines, although as respects the circle and a few other curvilinear figures, the properties of which have been specially investigated, so close an approach to the true area may be obtained that the error shall be less than the smallest quantity that can be assigned.
With regard to figures of irregularly curved outline, a general method has been devised of finding the area by means of “equidistant ordinates," which furnishes results of sufficient accuracy for all practical purposes. The application of this system is fully exemplified in the Distillery Instructions. It will only be requisite, therefore, in the present work to state briefly, a little further on, the principles on which the rule is founded.
A circle is a plane figure bounded by one continuous curved line called the circumference or periphery, as C D BA F P in the diagram. The distinctive property of the circle is, that all straight lines drawn from a certain point, E, within the figure to the circumference, are equal D to one another, and, therefore, the circumference is everywhere equally distant from this point, which is called the centre of the circle.
Any right line, as D P, passing through the centre, E, of the circle and terminated each way by the circumference, is called a diameter.
Any right line, such as D E, A E, PE, CE, drawn from the centre to the circumference, is called a radius, or semi-diameter.
An arc of a circle is any portion of the circumference, as B A F, DC, &c.
The chord or subtense of an arc, B A F, is the straight line B F joining the two extremities of the arc.
A semi-circle or half circle, is the figure contained by the diameter and the part of the circumference cut off by the diameter, as D A P, or D CP.
A quadrant, as A E D, is the figure contained by two radii which are perpendicular to each other, and the arc intercepted between them. A quadrant is the fourth part of a circle, or the half of a semi-circle.
* A plane figure bounded by a curved line, is not to be confounded with that which is properly termed a “ curved surface," such as the surface of a globe, no two points of which necessarily lie in the same plane.
A segment of a circle is the figuro contained by an arc and its chord, as B A F.
A sector of a circle is the figure contained by any two radii and the arc which they cut off, as D E C. When the containing radii are perpendicular to each other, as A E and E P, the sector thus formed is called a quadrant. It equal to the fourth part of the circle.
An ellipse, or ellipsis, is a plane figure bounded by a single curved line termed the periphery, and is such that if from any point, G, in the periphery straight lines, G F, G F be drawn respectively to two fixed points, F and F, within the figure, called the foci, the A sum of these lines will be always the
The middle point of the line F F which joins the foci, is called the centre of the ellipse, but it must not be supposed that, as in the case of the circle, all lines drawn from this centre to the periphery will be equal to one another.
Any straight line passing through the centre, and terminated each way by the curve, is said to be a diameter of the ellipse.
The diameter, A C, which passes through the foci, F and F, is called the transverse diameter, or major axis.
A straight line, B D, drawn through the centre perpendicular to the transverse diameter, is called the conjugate diameter or minor axis. Thus, an ellipse has two diameters or axes, the transverse and the conjugate, of which the transverse is always the longer.
An ordinate, H F, is any straight line, not passing through the centre, and perpendicular to either axis, connecting it with the periphery. When such a line is continued through the figure, it is called a double ordinate, and corresponds to a chord of a circle.
The section of a cone made by a plane which passes obliquely through its sides, is of the form of an ellipse.
A parabola is a curved figure, such as would be produced by cutting through a cone in a direction parallel to either of its slant sides. Several parabolas may be formed by different sections of the same cone, the bases and heights of which will all vary according to the distance of the cutting plane from the opposite parallel side of the cone. The line B D drawn to the middle point of the base, A C, is the axis or height of the parabola. The path of a cannon ball, or any other projectile, in the air, is nearly that of a parabolic curve. A jet of water issuing from the side of a vessel describes a similar curve. It is unnecessary for the purposes of this work to treat of the parabola or its properties, further than to give in the course of the following pages, the rule for computing the space included within the limits of a parabolic figure, such as A B C, and to show the ordinary method of describing the curve.
(9.) PRACTICAL GEOMETRY.—The only problems in practical geometry, with which it is essential that officers should be familiar, are the few following, it being assumed that the use of the common compasses in taking off the lengths of short lines and in drawing circles of any required radius, is fully understood by every person.
To bisect, or divide into two equal parts, a given straight line A B.
From the centres A and B, respectively, with any radius or opening of the compasses, greater than half A B, describe two arcs, cutting each other in C and D; draw C D, and it will cut A B in its middle point E. The theoretical method of bisecting a straight line is dependent on that of bisecting an angle. (See Euclid, Book I., Props. 9 & 10.)
A short line may also be bisected readily, by opening the compasses to about half the length of the line as nearly as can be judged. If upon applying this distance first from one extremity of the line and then from the other, the same point be reached by the extended leg of the compasses, the estimated distance is, of course, equal to the exact half of the line ; but should the leg of the compasses fall on different points, the small interval between these may be bisected by the eye, and the middle point of the line thus obtained.
At a given distance E, to draw a straight line C D, parallel to a given straight line 4 B.
From any two points m and r, in the line A B, with a distance equal to E, describe the arcs n and $ :-draw C D to touch these arcs, without cutting them, and it will be the parallel required.
Through a given point r to draw a straight line C D, parallel to a given straight line, A B.
From any point, n, in the line A B, with the distance n r, describe the arc r m ::-from centre p with the same radius,
ch describe the arc n 8 :-take the arc m r on the compasses, and apply it from n to 8 :-through r and s draw C D, which is the parallel required. The demonstration of this mode of drawing a parallel is contained in Euclid, Book I., Prop. 31.
From a given point P, in a straight line A B, to erect a perpendicular to that line.
1. When the point is in or near the middle of the line.
Note. This problem, as well as the following one, is usually performed by an instrument called the paralld ruler, which will be found in every case of geometrical instruments.
2. When the point P is at the end of the line.
With the centre, P, and any radius, describe the arc nrs:- from the point n, with the same radius, turn the compasses twice on the arc, as at r and 8 :--again, with centres r and s, describe arcs intersecting in C:draw C P, and it will be the perpendicular required.
Note.—This problem and the following one are usually done A with an instrument called the 8quare.
From a given point C to let fall a perpendicular upon a given line. 1. When the point is nearly opposite the middle of the line A B.
From C, as a centre, describe an arc to cut A B in m and n :-with centres m and n, and the same or any other radius, describe arcs intersecting in D:-through C and D
А draw C D, the perpendicular required.
2. When the point is nearly opposite the end of the line.
From C draw any line C m to meet B A, in any point, m :--bisect C m in n, and with the centre n, and radius C n, or m n, describe an arc cutting B A in P. Draw CP for the perpendicular required. (See Euclid, Book I., Prop. 12.)
To ascertain whether two given lines are perpendicular, or at right angles to each other.
Apply an accurately constructed square along the sides of the given angle, so that its vertex shall correspond with the corner of the square; then if the sides of the square and the angle coincide, the two given lines will be perpendicular to each other.
If a square cannot be procured at the time, draw any line, as C m, (last fig.) across the sides of the angle. Bisect it at n, and join n P. Then if, n P be equal to the half of C m, the angle at P is right; if otherwise, not. For, if C n be equal to the line joining n and P, the angle formed by that line with C P will be equal to the angle n C P. (Euclid I., 5.) And for a similar reason the angle n P m will be equal to n m P; that is, the entire angle at P will be equal to the sum of the angles at C and m. But, when one angle in a triangle is equal to the sum of the other two angles, that angle must be a right angle. (Euclid I., 32 Cor.)
Another mode of determining whether or not any proposed angle is a right angle, is suggested by Euclid I., 47:-Draw a line, as before, across the sides of the angle; measure the three sides of the triangle so formed, and find by calcu. lation whether the square of the length of the subtending side or hypotenuse is nearly equal to the sum of the squares of the sides containing the angle. But this is too troublesome to be much resorted to in practice.
To divide a given straight line (A B) into any number of equal parts.
Through A draw a straight line at any angle with A B, and through B draw B C parallel to A D. From the point A set off the required number of equal parts (in this illustration 6) on the line A D, which parts should be, as nearly as can be judged by the eye, of the same length as the required equal divisions of A B. Also, from B, set off on the line B C six equal parts of the same length. Join the points B and 0, 1 and 1, 2 and 2, &c. Then the points at which the lines so drawn cut A B, viz. a, b, c, d, e, will divide the given line A B, into the required number of equal parts. The principles of this operation are contained in Euclid I., 34, & VI. 2.
Note. The more obvious and simple way, when practicable, is to measure the length of the given line on a scale of equal parts, such as inches and tenths, or, if necessary, on a rule more minutely divided, and then by aid of the same scale to mark off the required number of equal fractions of the entire length.
Construction and use of the diagonal scale.
Draw a straight line, A B, of any a convenient length. Set off
any distance, A E, as a unit, according to the intended length of the scale. Repeat A E any number of times, as E G, G B, along A B. Draw C D parallel to A B, at any convenient distance ; then draw the perpendiculars, A C, EF, G H, B D, &c.
C 98765432ZF Divide A E and A C each into ten equal parts ; through 1, 2, 3, &c., draw lines parallel to A B, and through a, y, &c., draw the diagonal lines x F, y Z, &c., as in the annexed diagram.
A scale such as this, laid down on the surface of a slip of wood or ivory, is contained in every box of mathematical instruments. The object of the scale is to enable us to draw lines with exactness in any required proportion to one another, to lay down lines from a given measure, and to compare the lengths of different lines. The graduation of the logarithmic distances on the sliding rule is effected by means of an accurately divided diagonal scale.
Whatever length or number C F or A E is taken to represent, each of the equal divisions, F Z, Z 2, or E x, x y, &c., will be the one-tenth of it, and the subdivisions in the direction F E upwards, will be equal respectively to 1-100th, 2-100ths, &c., of the distance F C. Thus, if from C to F be reckoned 100, then from C to H is 200; from C to D is 300; also from F to 4 is 4-10ths of 100, that is, 40; from F to 7 is 70; from D to 8 is 280, and so on. If O F be assumed a unit or 1, then the divisions F Z, Z 2, &c., will be each one-tenth, and the divisions in the altitude from C upwards, or from E downwards, will be severally, 9-100ths, 8-100ths, &c., of unity.
The values of the sub-divisions in the vertical direction may be established as follows :-Let the distance 0 9 be taken as 1, and A
as 10, the
by the property of similar triangles, (Euclid, Book VI, Prop. 4,) A C (10) is to 09 (1) as a 9 (1) is to to Again, A C (10) : C 9 (1) :: A8 (2) : * ; and similarly for each of