solid geometry. Parts of the eleventh and twelfth books are still read in our universities.

During the dark ages 'The Elements of Euclid' were not studied in Europe. The geometry then taught consisted merely of a few definitions and axioms, without any attempt at mathematical demonstrations. The Saracen conquerors of Alexandria, however, became familiar with the Elements,' and translated it into Arabic, and from Arabic manuscripts what we may consider as the first modern edition of Euclid was translated by Adelard of Bath, an English traveller, who, at some period of the twelfth century, visited the schools of Cordova and Bagdad. The first printed 'Euclid,' which was published at Venice in 1482, was also a translation from the Arabic. The first translation from Greek manuscripts was in 1505. These were all translations into the Latin language, which was throughout the middle ages the universal language of scholars. In Queen Elizabeth's reign Billingsley, a citizen of London, translated the 'Elements into English, and from his time to our own numerous editions and commentaries have continually issued from the press, while, at the same time, from the revival of letters to the present day, Euclid's work has been studied in the schools and universities of Europe.

Next, before we begin to study the book whose history we have briefly surveyed, let us, in order that we may pursue our subject intelligently, consider what ends we have in view in learning this science of geometry. The advantages are, I think, twofold—direct and indirect. First, geometry teaches us, as we have already seen, the laws which regulate the relations of the parts to the whole in simple figures, and of these figures to each other. Now, if we look around us, all the objects which meet our glance are either modifications of, or combinations of the simple figures, and having studied the proportions of the elementary figures in the abstract we shall be able to apply them to all the surrounding objects of life, and we shall know the fundamental principles upon which painting, architecture, and engineering depend; also on these same

principles depend the useful and the beautiful in furnishing and adorning our houses, in laying out our gardens, and even in designing our dress.

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Again, the study of geometry tends to develop and cultivate some of our most valuable mental qualities. By it the power of reasoning is continually exercised. Euclid does not call upon us to believe or to take on authority anything but a few self-evident truths. Every assertion that he makes he afterwards proves. And, in order to comprehend his reasoning, we must be careful to follow the argument closely, or, by a moment's inattention, a point will escape our notice, and we shall find at the conclusion a flaw in the demonstration which will make it valueless, and oblige us to retrace our steps to the very commencement. This power of fixing the attention, or of concentrating our ideas, is most necessary to our ever succeeding in attaining any great or worthy object.

We have now traced the origin of the science of geometry, and the history of its development, chiefly under the influence of Euclid's genius. We have considered, also, some of the reasons for studying it. We shall, in our next lesson, proceed to the consideration of the first book of the 'Elements.'

LECTURE II.

ON THE DEFINITIONS, AXIOMS, AND POSTULATES.

OUR lesson to-day will embrace (1) the definitions in which Euclid gives the precise meaning of the terms he intends to employ; (2) the postulates in which he asks to be allowed to do three simple things, and to take for granted three truths belonging to geometry alone; and (3) the axioms, a list of nine truths, which, by the nature of our minds, we perceive intuitively, and which are common to all magnitudes to which they can be applied.

First, the definitions, 35 in number (in the original öpot, boundaries); these, in order to assist our memory, we may classify as follows:

Nos. 1-7 are definitions of lines and surfaces.

Nos. 15-19 define the circle and its parts.

OF LINES AND SURFACES.

1. A point is that which has no parts or magnitude.'

2. A line is length without breadth.'

3. The extremities of a line are points.'

4. A straight line is that which lies evenly between its extreme points.'

5. A superficies or surface is that which has only length and breadth.'

6. The extremities of a superficies are lines.'

7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.' (This is the carpenter's test, trying a flat surface with a straight edge.)

The ideas which are formulated in these definitions have been familiar to us, though perhaps unconsciously, from early childhood. The uncut end of the pencil I hold in my hand, the outside of the cover of the book on the table, or any of its pages, are plane surfaces (def. 7). The polished wood of the pencil forms a rounded or curved surface; both kinds of surface are included in def. 5. Again, the edges of the book are lines (def. 6) having length without breadth (def. 2). Any two edges of the book meet in a point (def. 3) which obviously has no size or magnitude (def. 1).

To these definitions an objection is frequently urged which we must briefly examine. It is said that defs. 1, 2, and 5 are incorrect, since any surface must have some thickness, any line some breadth, and any point some size, however small. We have seen that the outside of the book-cover is a plane surface. Now, suppose we wish to measure this surface in order to find out what space on the table it would occupy; you will tell me that the book is so many inches broad (say 6 in.), and so many inches long (say 9 in.), and that it will therefore occupy a space of so many (say 54) square inches. Now, if I ask, but how thick a space will it occupy? will you not be inclined to smile at such a question and answer, 'The surface of the book contains 54 sq. in., and will therefore cover 54 sq. in. of the surface of the table, but it has nothing to do with thickness, either that of the book or that of the table.' Hence, it is evident that we have a clear conception of surface as possessing the properties of length and breadth without thickness, and that Euclid's definition is strictly true.

Again, if I ask you to measure an edge of the book (which we took as our example of a line), you will tell me it is so many inches long; but if I ask how wide it is, the ques

tion strikes you as absurd, it has no width, nor could you measure the point at which any two edges meet at all, and it consequently cannot be said to have any size whatever. We must then admit that these definitions give perfectly accurate descriptions of the ideas which we have in our minds, and express by the words surface, line, and point. In other words, we may say that a point indicates position; a line, distance from one point to another; a surface, the extension of any body in one plane, i.e. the amount of space it would occupy in one plane surface or area.

But I wish you also to notice that the lines and surfaces with which we are familiar from experience exist always in connection with each other, and also with what we have seen is the third geometrical property of matter, viz. thickness or depth, i.e. the lines are boundaries of some surface, the surfaces of some solid body. But in that part of the subject which we are now beginning Euclid requires us to think of surfaces and lines as if they existed separately, just as in comparing colours we should speak of white and red, without any reference to the thing which was white or red. The marks made by a pen or pencil to represent lines are very thin and narrow solids of ink or black lead.

OF ANGLES.

We pass on to the simplest combination of lines, i.e. two lines meeting in a point forming a corner or angle (Lat. angulus, a corner).

Def. 8. ‘A plane angle is the inclination of two lines to one another in a plane, which meet together but are not in the same direction.'

9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together but are not in the same straight line.'

Def. 8 includes angles formed by two curved lines, or a straight and a curved line. With these we shall not be concerned. The word rectilineal simply