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Euclid contain as much real geometrical study as three or four times that number in some other systems that might be mentioned.
It must also be remembered that Euclid's method is not identical with Euclid's text. The former may be retained in all its essential features, while the latter may have its phraseology or arrangement modified, so as to make the course of the reasoning more evident to a beginner. This is what has been attempted in the present work. The author has endeavoured to enable the learner to gain an accurate idea of the chain of reasoning in each proposition, to get him to discriminate clearly between the data and the inferences, to point out the logical pitfalls that he must guard against, and, generally, to present the propositions in the shape which would probably be given to them by an intelligent teacher, who was endeavouring to make a beginner understand the text of Euclid. It requires some experience in teaching and examining boys to become aware in how many cases learners who have “gone through Euclid,” and can write out the propositions with perfect exactness, are absolutely destitute of any intelligent comprehension of the reasoning. The result achieved is a mere effort of memory. Over and over again the author has met with those who could write out or go through the fifth proposition of the first book, as it stands in the text of Euclid, without a mistake, and yet had not even the slightest idea that the enunciation contained a statement of something that was known to begin with, and something else which had yet to be deduced from the data.
By way of introduction to each proposition, the various axioms, postulates, and propositions which are assumed in the course of the construction or demonstration are stated at length. Many may think this superfluous; but it will usually be found that when boys are set to prepare propositions from the text of the ordinary editions of Euclid, not more than one in twenty gives himself the trouble to recollect, or refer to the previous propositions which are employed in any given demonstration. By setting forth explicitly and at length all that is assumed as known in the course of a demonstration, the mutual dependence of the propositions is far more clearly perceived, and each proposition becomes an independent whole, admitting of being comprehended and mastered without a knowledge of the mode in which the subordinatu propositions are to be established.
In teaching geometry I attach very great importance to the repetition of the propositions with figures to different shapes and in different positions. Another point which should be rigidly insisted on is, that in all cases in which construction is required, the illustrativo diagram should be drawn step by step according to the directions given. In the definition of angular magnitude some eminent teachers avail themselves of the conception of the revolution of a straight line from a given position round a fixed point in the line itself. This furnishes the beginner with an admirable illustration, but I conceive it is a mistake to embody it in a definition, since the conception of an angle may be formed independently of all idea of motion or revolution; just as it would be unnecessary (not to say incorrect) to define a line as produced by the motion of a point. In fact it is a mistake to say that the motion of a point produces a line. All that it produces is a change in the position of the point. The line along which the point has moved is a separate creation of the mind. In like manner the revolution of a line does not produce angular magnitude; it produces only a change in the position of the line.
Angular magnitude is a separate product of thought, for which a certain relative position (not motion) of two lines is essential.
Whatever may be the difficulties or defects of Euclid's “
· Elements," it is tolerably certain that it will be a long time before it is superseded as a textbook. Meanwhile it is of importance that those who have to use the book should understand it as thoroughly as possible. It is hoped that the present treatise may help many towards the attainment of that object.
The greater part of this work is a second edition of “The First Book of Euclid explained to Beginners;” to this has been added an investigation of the Second Book, conducted on similar principles.
C. P. MASON.
FIRST BOOK OF EUCLID
EXPLAINED TO BEGINNERS.
THE First Book of Euclid treats of the properties of figures made up of points, straight lines, and circles, and drawn in one and the same plane.
DEFINITIONS. Def. I. A point is something which is indivisible, that is, which has no magnitude.
Def. II. A line is that which has length without breadth.
Def. V. A superficies, or surface, is that which has length and breadth, but not thickness.
The use of these definitions is to caution us that in geometry we are not reasoning about such figures as we can draw with a pen or pencil upon a sheet of paper. Figures of this kind are used in the study of geometry, but only for the purpose of helping the memory and imagination. The propositions of geometry can only be absolutely true of figures that are absolutely perfect. But absolutely perfect figures can only be imagined. It is impossible even to get a continuous surface. The only physical surfaces that we can deal with are made up of atoms lying very close to each other, but not absolutely in contact. The finest stroke that is visible must have a certain amount of breadth, or it could not reflect light or be visible at all; and moreover, no line that we can draw
is without break, for it must be drawn upon some surface, and no visible surface is continuous. The smallest dot that can be made is a spot of some size. But these rude material surfaces, lines, and points suggest to our minds the perfect surfaces, lines, and points which we deal with in geometry. The definitions of these given by Euclid are entirely negative in their character, and warn us that we must correct our first rough notion of a point till, by an effort of the mind, we have got rid of the idea of size altogether, and in like manner correct our first rough notion of a line till we have got rid of the idea of breadth. The use of the figures drawn in geometry is to aid us in imagining those perfect figures about which we really reason.
As the surfaces and lines treated of in geometry exist only in our conceptions, and are not actual material objects, we can speak of surfaces and lines of infinite extent, that is, having no limits or termination.
Def. III. The extremities of a finite or bounded line are points.
Def. VI. The extremities of a finite or bounded surface are lines.
Def. IV. A right (or straight) line is one which lies evenly between its extremities.
It is impossible to define the term right or straight withont using some equivalent word which needs definition quite as much as right or straight itself. Our notion of straight is one of those primary conceptions of form which cannot be analysed into simpler elements. Euclid's definition is, moreover, faulty, inasmuch as it would not be applicable to a straight line of infinite length, which has no extremities.
Def. VII. A plane surface is one which lies evenly between its extremities.
This definition errs in the same manner as the last. It is better to define a plane surface as a surface such, that if any two points be taken upon it, the straight line drawn from the one point to the other lies entirely on the surface.