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It is easy to foresee that after the step marked 4 (Ex. 1) is reached, the work of the subsequent steps can have little or no influence upon the three leading figures 6·23 of the forthcoming divisors, so that regarding these three figures as constant, and recognising the others, 8092, only for the sake of what is carried from them, we may, as above, make sure of at least three true decimals of the root, beyond the three already found, by common division, provided we reject a figure of the constant divisor, 6·23, at each step, taking care to secure accuracy in the carryings from the rejected figures.

When you arrive at the Theory of Equations, you will find a systematic method of applying such contractions fully explained, as also a complete investigation of the principles on which the foregoing practical operations depend: what is here exhibited is only a small isolated portion of a department of modern Algebra of very comprehensive scope, and of as much theoretical interest as of practical utility. Ex. 1. The cube-root of x6 + 9x5 + 6x4 x2+3x-7.

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15x3y + 69x1y2 - 138x2y1 — 60.xy3 — 8x is x2 -5xy -2y2

I here conclude the treatise on Elementary Algebra. The subject in its widest acceptation is one of very considerable extent—I might almost say of unlimited extent; as there are no definite bounds to its operations. In the preceding treatise, my object has been to unfold to you, fully and perspicuously, the leading principles of the science; and thus to lay a sufficiently secure basis for future researches. There is one department of the subject-the general theory of Logarithms and Series-which I have not touched upon here. It is a part of Algebra which is marked by peculiar features, and is occupied with investigations different in kind and in object from those necessary for the solution of an algebraical equation, or for the reduction of an algebraical expression; and is, moreover, of sufficient importance to merit distinct consideration.

J. R. YOUNG.

PLANES :

BEING THE FIRST TWENTY-ONE PROPOSITIONS OF THE ELEVENTH BOOK OF

EUCLID'S GEOMETRY.

[The Elements of Euclid, given with considerable detail in a former treatise, were limited to the subject of plane Geometry, as this furnishes basis sufficient for plane Trigonometry. The Geometry of Planes and Solids was intended to have been postponed till our treatise on Spherical Geometry rendered its introduction necessary. But as the consideration of Mechanical Forces in Space, and the science of Crystalography will very shortly appear in another department of the “CIRCLE,” we have thought it advisable to introduce Solid Geometry in this place. Moreover, several of the subjects included in this volume will be more simply treated, and more easily understood, if a knowledge of some of the elementary properties of planes and straight lines be previously acquired. For instance, the calculation of the solid contents of an earthwork will be best given in the Treatise on Mensuration ; but this calculation, of course, depends on the relations of solid space. Again, we propose to present the reader with a short treatise on linear perspective-a subject which, if treated as a science, and not merely as an art, also involves a knowledge of the elementary relations of solid space : the subject of Spherical Trigonometry, likewise, is best given in the present volume; and this science, as its name denotes, treating of triangles which are described on the surface of a sphere, cannot be taught without reference both to the properties of planes and to the elementary propositions of Spherical Geometry.

The treatise on planes contains the first twenty-one propositions of the Eleventh Book of Euclid's Geometry, with so many definitions as are requisite to enable the student fully to understand them; and the Author has himself explained the object he had in view in drawing up the Treatise on Spherical Geometry.—THE EDITOR.]

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Introduction.-The figures, lines, angles, &c., the properties of which form the subject of the First Six Books of Eulcid’s Geometry, are supposed to lie in one plane, * i. e. of length or to be in space of two dimensions.* The following treatise contains

and breadth, few elementary propositions on the relations between lines, angles, &c., which do not lie in one plane, but are in solid space,--or space of three dimenti. e. of length,

sions.t The student will find the following propositions very easy, breadth, and when once he has distinctly conceived the meaning of their enunciations. The

figures which are given to each proposition cannot represent the proposition to the eye so perfectly as in the former books, in consequence of their having to be drawn in perspective. It is hoped, however, that the shading introduced into the diagrams will aid the student in conceiving the proposition they belong to.

It is to be added, that, as in Plane Geometry, we are allowed to draw lines in any direction, and to produce them to any extent, so in solid Geometry we are allowed to draw planes in any direction, and to produce them to any extent. Moreover, two lines intersect in a point; in like manner it will be shown that two planes intersect in a line. Also, as we may suppose,a line to revolve round a point till it comes to a point on its plane, so we may suppose a plane to revolve round a given line until it comes to a given point situated anywhere in space.

MATHEMATICAL SCIENCES.-No. VIII,

R

DEFINITIONS.

I.

A straight line is perpendicular, or at right angles to a plane, when it makes right angles with every straight line in that plane which meets it.

Thus, if BD be a plane, PA a line perpendicular to it. Through A draw any lines AB, AC, AD.... in that plane, then will PAB, PAC, PAD, &c., be right angles.

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II.

A plane is perpendicular to a plane, when a straight line drawn in one plane perpendicular to the intersection of the planes is at right angles to the other plane.

Thus, let ABD, ABC, be two planes, let the former be perpendicular to the latter, and let AB be the line of intersection of the planes in the plane ABD, draw PN at right angles to AB. Then is PN at right angles to the plane ABC.

WB

III.

P

The inclination of a straight line to a plane, is the acute angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane drawn from any point of the first line above the plane meets the same plane.

Thus, let ANB be a plane, AP a line meeting the plane in A; from P draw PN perpendicular to the plane, and meeting the plane in N. Join AN, then the angle PAN is the inclination of the line PA to the plane ANB.

IV.

The inclination of a plane to a plane, is the acute angle .contained by two straight lines drawn from any one point of their common section at right angles to it, one upon one plane, the other upon the other.

Let PAC, PBC be two planes intersecting in the line PC. From P in the former plane, draw PA at right angles to PC; and from the same point P on the latter plane, draw PB at right angles to PC. Then, if BPA be an acute angle, this is the inclination of the planes to each other.

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V.

Two planes have the same inclination to one another which two other planes have, when the said gles of inclination are equal to one another.

VI.

Parallel planes are such as do not intersect, though produced ever so far in all directions,

VII. A solid angle is that which is made by the meeting of more than two plane angles, which are not in the same plane, in one point.

PROPOSITION 1.-THEOREM.

One part (AB) of a straight line (ABC) cannot be in a plane, and another part (BC)

be above it.

For let us suppose this possible, then since the straight line AB is in the plane, it can be produced in that plane : let it be produced to D. Now, suppose a plane to pass through the straight line AD, and let it be turned round that line, till it comes to the point C. Then because B and C are in the plane, the straight line BC is in it:* :. there

* 6 Def. I. are two straight lines ABC, ABD in the same plane, having a common + Cor. 11 I. segment AB, which is impossible.† Q. E. D.

PROPOSITION II.-THEOREM.

Two straight lines (AB, CD) which cut one another (in the point E) are in one plane. And

three straight lines (BC, CE, EB) which meet one another, are in one plane. Let any plane pass through EB, and let the plane be turned about EB produced if necessary, until it pass through the point C. Then because the

* 6 Def. I. points C and E are in this plane, the line CE is in it.* For the same reason the straight line BC is in the same plane, and by the hypothesis EB is in it; .;; the three straight lines BC, CE, EB are in one plane. But AB is in the same plane as EB, and DC

#1 XI. as EC. Also AB and DC are in the same plane. Q. E. D.

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PROPOSITION III.-THEOREM,

If two planes (AB, BC) cut one another, their common section (DB) is a straight line.

For, if not, since D, B are points in the plane AB, draw the straight line DFB in that plane, and similarly draw the line DEB in the plane BC. Then because these two straight lines have the same extremities, they enclose a

* 10 Ax. I. space, which is absurd; ;. the common section BD cannot but be a straight line. Q. E. D.

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