Euclid's work comprises thirteen books, of which the first four and the sixth treat of Plane Geometry; the fifth, of the Theory of Proportion, applicable to magnitude in general; the seventh, eighth, and ninth, are on Arithmetic; the tenth, on the Arithmetical Characteristics of the divisions of a straight line; the eleventh and twelfth, on the Elements of Solids; and the thirteenth, on the Regular Solids. To the thirteen books by Euclid, Hypsicles of Alexandria, about A. D. 170, added the fourteenth and fifteenth books, also on the Regular or Platonic Solids.

In modern times it is not usual to read more than six books of Euclid's Elements. The seventh, eighth, ninth, and tenth books treat of Arithmetic and of the Doctrine of Incommensurables, and have no proper connection with the first six books; and the eleventh and twelfth books, comprehending the First Principles of Solid Geometry, are to a considerable degree superseded by other Treatises.

Of the Six Books, the first may be described in general terms as treating of the Geometry of Plane Triangles; the second, of Rectangles upon the parts into which a straight line may be divided; the third book, of those Properties of the Circle which can be deduced from the preceding books; the fourth book, of such regular and straight-lined figures as can be described in or about a circle; the fifth, of Proportion with regard to magnitude in general; and the sixth, of similar figures, and of Proportion as applied to Geometry.

Our proposed limits confine us, for the present at least, to the first and second books. The first book, besides the Definitions, Postulates, and Axioms, contains forty-eight propositions, of which fourteen are problems for giving power to construct various lines, angles, and figures; and thirty-four are theorems, being the expositions of new geometrical truths. Of these theorems, some may be regarded as simply subsidiary to the proof of others more important, and of wider and more general application. The Propositions to be ranked among those of high importance, are Props. 4, 8, and 26, containing the criteria of the equality of triangles: Prop. 32 proving that the three interior angles of every triangle are together equal to two right angles: Prop. 41 declaring that a parallelogram on the same base and of the same altitude as a triangle, is double of the triangle: and Prop. 47 demonstrating that the square on the hypotenuse of a

right-angled triangle, is equal to the sum of the squares on the base and perpendicular.

The second book treats of the properties of RIGHT-ANGLED PARALLELOGRAMS, contained by the parts of divided straight lines. There are fourteen Propositions; of which Props. 11 and 14 are problems,—the other twelve are theorems. Props. 12 and 13 give the Elements of Trigonometrical Analysis, or the Arithmetic of sines, and are of great use in the Higher Geometry: the other propositions may be classified according to the mode of dividing the line or lines; Prop. 1 relating to the rectangles formed by one undivided line and the parts of a divided line; Props. 2, 3, 4, 7, and 8, to the rectangles formed by a line and any two parts into which it may be divided; Props. 5 and 9, to the rectangles on a line divided equally and unequally; and Props. 6 and 10, to the rectangles formed on a line bisected and produced.

The English Translation of Euclid, published by Dr. Robert Simson, of Glasgow, in 1756, has nearly, in some form or other, superseded all others, and is considered the standard text of an English Euclid. As containing "the Elements of Geometry," it is "unexceptionable, but is not calculated to give the scholar a proper idea of the Elements of Euclid," as Euclid himself left them. Various alterations, additions, and improvements, were made by Simson: but "with the exception of the editorial fancy about the perfect restoration of Euclid, there is little to object to in this celebrated edition. It might indeed have been expected that some notice would have been taken of various points on which Euclid has evidently fallen short of that formality of rigour which is tacitly claimed for him. We prefer," says De Morgan, "this edition very much to many which have been fashioned upon it,-particularly to those which have introduced algebraical symbols into the demonstrations in such a manner as to confuse geometrical demonstration with algebraical demonstration."-(See the Article, Eucleides, by De Morgan, in Smith's Dictionary of Greek and Roman Biography, Vol. II., pp. 63-74.)

In the face of such authority, it may seem bold to advocate the use of a Symbolical Notation; but, within certain limitations, the symbols of Arithmetic and of Algebra have a universal meaning, and may therefore be employed without any disadvantage, and certainly without confusion in our ideas. The pre

caution needed, is, that we take care not to depart from the strictly geometrical application.

For an outline of the origin and progress of the science of Geometry, the learner should consult the Introduction to the Elements of Euclid, edited by Robert Potts, M.A., Trinity College, Cambridge.

SYMBOLICAL NOTATION AND

SECTION II.

ABBREVIATIONS THAT MAY BE USED.

I.-Signs common to Arithmetic, Algebra, and Geometry.

* When an s is added to the sign, the plural is denoted.

A single capital letter, as A, or B, denotes the point A, or the point B.

Two capital letters, as AB, or CD, denote the straight line A B, or CD.

2

Two capital letters, with the figure just above to the right hand, as A B2, denote, not the square of À B, but the square on the line A B.

Capital letters, with a point between them, as AB.CD, denote, not the product of A B multiplied by CD, but the rectangle formed by two of its sides meeting in a common point.

Q.E.D., quod erat demonstrandum, which was the thing to be proved. Q.E.F., quod erat faciendum, which was the thing to be done.

SECTION III.

EXPLANATION OF SOME GEOMETRICAL TERMS.

A Definition is a short description of a thing by such of its properties as serve to distinguish it from all other things of the same kind.

A Postulate is a self-evident problem, the admission of which is demanded without formal proof.

An Axiom is a self-evident theorem, or the assertion of a truth, which does not need demonstration;—it is worthy of credit as soon as stated.

A Proposition is something proposed to be done, as a problem; or to be proved, as a theorem.

A Problem is a proposal to do a thing, to construct a figure, or to solve a question.

A Theorem is the assertion of a geometrical truth, and requires demonstration.

it.

The Data are the things granted in a problem;

The Quæsita are the things sought for in it;

The Hypothesis is the supposition made in a theorem ;

The Conclusion is the consequence or inference deduced from

The General Enunciation of a proposition sets forth in general terms the conditions of the problem, or theorem, with what has to be done, or with what is inferred or concluded.

The Exposition, or Particular Enunciation, sets forth the same conditions with an especial reference to a figure that has been drawn.

The Solution of a problem shows how the thing proposed may be done.

The Construction prepares, by the drawing of lines, &c., for the demonstration of a proposition.

The Demonstration proves that the process indicated in the solution is sound, or that the conclusion deduced from an hypothesis, is true; i. e., in accordance with geometrical principles.

The Recapitulation, or Conclusion, 'is simply the repetition of the proposition, or general enunciation, as a fact, or as a truth, with the declaration Q.E.F., or Q.E.D.