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to two right angles, when his father discovered what he was doing. This, be it remembered, was when he was only twelve years old

Passing over Huygens and Gregory, we pause a moment to admire the enthusiasm of Dr. Barrow, the illustrious preceptor of Newton, Though educated for a theologian, geometry had attractions which he could not resist. He, like Plato, considered the contemplation of it, as not unworthy of the deity, and inscribed the edition which he published of Apollonius, with these words : “God himself geometrizes; O Lord, how great a geometer thou art !"

It would seem that few discoveries now remained to be made in geometry. The labours of the eighteenth century were chiefly directed to the extending of its applications, thus making it the instrument instead of the object of discovery: Or if any still attempted to improve the science itself, it was by remodelling its elements, and not by adding to their number. To this class belong Simson, Playfair, and Legendre. We might mention many others, but we limit ourselves to these, because they are the authors chiefly studied in the United States. Simson and Playfair, two Scottish professors, have each published improved editions of Euclid, which leave little to be desired on the subject of elementary geometry, according to the ancient or Euclidean method. Legendre, the most eminent of French geometers, has produced a work, which deservedly stands at the head of modern systems. It has been many times translated, and has passed through a great number of editions. The translation which is chiefly studied in this country, was executed by Professor Farrar of Harvard University. Respecting these three works, we shall only add, that those who would understand geometry as it was left by Euclid, must study Simson; those who would unite modern improvements with the rigid method of the ancients, must study Playfair; and those who would have a complete view of geometry as it now is, without particular regard to the ancient method, must study Legendre.

As the student may desire to know in what respects the ancient and modern methods differ, we shall briefly state their general characteristics. Both agree in this, that certain principles or truths are taken for granted to begin with. They are taken for granted, because they cannot be proved; being self-evident the moment they are stated. These

are called axioms, and are to geometry, what the foundations are to a building. Euclid's axioms are the following:

1. Things which are equal to the same are equal to one another.

2. If equals be added to equals the wholes are equal.

3. If equals be taken from equals, the remainders are equal.

4. If equals be added to unequals, the wholes are unequal.

5. If equals be taken from unequals, the remainders are unequal.

6. Things which are double of the same, are equal to one another.

7. Things which are halves of the same, are equal to one another.

8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.

9. The whole is greater than its part.
10. Two straight lines cannot inclose a space.
11. All right angles are equal.

12. If a straight line meets two straight lines so as to make the two interior angles on the same side of it taken together, less than two right angles, these straight lines being continually produced, shall at length meet upon that side upon which are the angles which are less than two right angles.

The last of these has been added by Euclid's Commentators.

The two methods differ in this. Euclid never supposes a line to be drawn, until he has first demonstrated the possibility and pointed out the manner of drawing it. But in three cases the possibility cannot be demonstrated, because it is self-evident. These cases are called postulates, and are the following:

1. Let it be granted that a straight line may be drawn from any one point to any other point.

2. Let it be granted that a terminated straight line may be produced to any length in a straight line.

3. Let it be granted that a circle may be described from any centre, at any distance from that centre.

The moderns, as Legendre, for example, are not thus scrupulous; but constantly suppose lines to be drawn,

without demonstrating the possibility or explaining the manner.

Lastly, the two methods differ in this. The moderns avail themselves of all the aid which Algebra can afford them. The ancients were ubacquainted with Algebra. Accordingly Euclid was obliged to demonstrate the laws of proportion geometrically. Whereas in modern systems, these laws are supposed to have been previously demonstrated by the help of Algebra. The moderns also derive great advantage, in every part of geometry, from the use of Ålgebraic signs and symbols. The ancient reasonings, for want of these, were rendered exceedingly cumbrous and circuitous.

These are some of the general distinctions. But the student who would be able to estimate the comparative merits of the two systems, must examine both for himself.

ERRATA.

Page 4 Line27 for A. B. to C. D. read A B to CD

8 s vertices and centres vertices as centres 10 37 A E CEB C D

A EC=BED 12 43 intersection

intersection.
31 11 G=C

G=B
38 11 perimter
42 40 centre 0

centre N
41 prism is a circle

prism is a cube

perimeter

66 60

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ELEMENTS OF GEOMETRY,

SECTION FIRST.

Of Lines and their Relations.

1. The study of geometry properly begins with the consideration of a point, this being the first and simplest geometrical idea. If you were required to make a point with a pencil upon paper; you would merely place the sharpened end upon the paper, without moving it in any direction. If the pencil be as sharp as possible, this is the nearest approach you can make to a geometrical point, which is defined to be-position merely, without any magnitude- But as you cannot represent to the eye that which has absolutely no extension, it is sufficiently near the truth to call a point-that which has an infinitely small extension—. By infinitely small, we mean for the present, the smallest that can possibly be conceived.

2. A point is the beginning and end of a line: for if you were required to make a line you would begin by placing the point of your pencil upon the paper; you would proceed to move it along the surface of the paper; and you would end by ceasing to move it. Here you make one point by placing the pencil; you make a line by moving it; and you make another point where you cease to move it. Accordingly we say--a line is the path described by the motion of a point; and—the boundaries of a line are points. It is evident that if the describing point had no extension, the line would only have that which it acquires from the motion, namely length, without any breadth or thickness. But as such a line could not be represented to the eye, it is sufficiently near the truth to sayma line has ** length, with only an infinitely small breadth and thickness— 3. You are next to form an idea of a straight line. This

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