INTRODUCTION.

SECTION 1.

GRADUAL GROWTH OF GEOMETRY AND OF THE ELEMENTS OF

EUCLID.

GEOMETRY, land-measuring, as the word denotes (from gee, earth or land, and metron, a measure), was in its origin an Art, and not a Science it embraced a system of rules, more or less complete, for performing the simpler operations of land-surveying; but these rules rested on no regularly-demonstrated principles, they were the offspring rather of experiment and individual skill, than of scientific research. In the same way poems-even some of the noblest-were composed before the principles of poetry had been collected into a system; languages were spoken, long before a grammar had been compiled; and men reasoned and debated before they possessed either a logic or a rhetoric: so measurements were made, while as yet there was no accurate theory of measuring-no abstract speculations concerning space and its properties.

The points and lines of such a Geometry were necessarily visible quantities. A mark, which men could see, would be their point; a measuring-rod, or string, which they could handle, their line; a wall, or a hedge, or a mound of earth, their boundary. The first advance beyond this would be to identify the instruments which they used in measuring, with the lines and boundaries themselves : the finger's breadth, the cubit, the foot, and the pace, would become representatives of a certain length without reference to the shape. It was only as the ideas and perceptions of those who cultivated the art of measuring grew more refined and subtile, that an Abstract Geometry would be evolved, such as Mathematicians understand by the term, in which a point marks only position; a line, extension from point to point; and a surface, space enclosed by mathematical lines.

Geometry, thus understood, has been defined in general terms to be "the Science of Space," or "the Science of Form." It

investigates the properties of lines, surfaces, and solids, and the relations which exist between them. Plane Geometry investigates the properties of space under the two aspects of length and breadth; Solid Geometry, under the three-of length, breadth, and thickness. It is the consideration of the Elements of Plane Geometry on which we are about to enter.

The Truths of Geometry, as a science, regularly as they are laid down and deduced in the Elements of Euclid, were not worked out by one mind, nor established in any systematic order. Some were discovered in one age, some in another; two or three propositions by one philosopher, and two or three by some one else. The collection of geometrical truths had thus a gradual growth, until it received comparative completion at the hands of Euclid of Alexandria.

Thales, who predicted the eclipse that happened B.C. 609, is said to have brought Geometry from Egypt, and to have established by demonstration Propositions 5, 15, and 26, of bk. i.; 31, iii.; and 2, 3, 4, and 5, of bk. iv. Pythagoras, born about 570 B.C., was the first who gave to Geometry a scientific form, and discovered Propositions 32 and 47 of bk. i.; Oenopides, a follower of Pythagoras, added the 12th and 23rd of bk. i.; and Eudoxas, B.C. 366, a friend of Plato, wrote the doctrine of proportion as developed in the fifth book of the Elements. These assertions may not rest on the firmest authority, yet they show, even if they are only surmises, that Geometry was regarded by the Greeks as a science of very gradual formation, receiving accessions from age to age, and from various countries. It was at first a set of rules, until philosophy investigated the principles on which the rules were founded, and out of the chaos created knowledge.

According to Proclus, EUCLID of Alexandria flourished in the reign of the first Ptolemy, B.C. 323-283. To him belongs the glory, for such it is, of having collected into a well-arranged system the scattered principles and truths of Geometry, and of having produced a work, which, after standing the test of above twenty centuries, seems destined to remain the Standard Geometry for ages to come.

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 that are 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; 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. It is on these Propositions of the first book-namely, 4, 8, and 26, 32, 41, and 47 that Geometry in its after applications mainly depends, and, therefore, must they be most thoroughly understood and mastered.

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 itparticularly 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

*See Lacroix' Essais, ed. 1838, p. 227. "L'histoire des Mathématiques prouve aussi que c'est l'usage de plus en plus étendu des symboles arbitraires imaginés dans la vue d'abréger les expressions ou de mettre en évidence leur analogie, qui a contribué le plus à l'avancement de la science; en soulageant la mémoire et facilitant les combinaisons des relations données et des raisonnemens."

certainly without confusion in our ideas. The precaution 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.

SECTION II.

SYMBOLICAL NOTATION AND ABBREVIATIONS THAT MAY BE USED.

Signs in Geometry possess an accuracy equal to that of words, and far greater clearness than the undivided paragraphs of Simson, or even than the demonstrations by Potts, so carefully broken up into clauses. Cambridge may not approve the use of certain symbols in Geometry which once she allowed, but Lacroix, in a passage just quoted, testifies to their advantage: thus fortified, we make no further apology for using them, especially after declaring their meanings.

One strong recommendation of them has thus been pointed out: "It is quite possible, and, in fact, frequently happens, that a boy gets up his Euclid lesson by rote, from the ordinary editions now in use;" but it is almost, if not quite, impossible to bring the memory simply into play when a proposition of Euclid is set forth in symbolical language: the reasoning powers must then be exercised, or the work will not be accomplished.

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