Given two angles A and B, and the side a, opposite to one of them. To find b, the side opposite to the other.

The other two cases, when the three sides are given to find the angles, or when the three, angles are given to find the sides, are resolved by the 29th, (the first of NAPIER'S Propositions,) in the same way as in the table already given for the case of the oblique angled triangle.

There is a solution of the case of the three sides being given, which it is often very convenient to use, and which is set down here, though the proposition on which it depends has not been demonstrated.

Let a, b, c, be the three given sides, to find the angle A, contained between b and c.

In like manner, if the three angles, A, B, C are given to find c, the side between A and B.

:

These theorems, on account of the facility with which Logarithms are applied to them, are the most convenient of any for resolving the two cases to which they refer. When A is a very obtuse angle, the second theorem, which gives the value of the cosine of its half, is to be used; otherwise the first theorem, giving the value of the sine of its half its preferable. The same is to be observed with respect to the side c, the reason of which was explained, Plane Trig. Schol.

NOTES

ON THE

FIRST BOOK OF THE ELEMENTS.

DEFINITIONS.

I.

In the definitions a few changes have been made, of which it is necessary to give some account. One of these changes respects the first definition, that of a point, which Euclid has said to be, That which has no parts, or which has no magnitude.' Now, it has been objected to this definition, that it contains only a negative, and that it is not convertible, as every good definition ought certainly to be. That it is not convertible is evident, for though every point is unextended, or without magnitude, yet every thing unextended or without magnitude, is not a point. To this it is impossible to reply, and therefore it becomes necessary to change the definition altogether, which is accordingly done here, a point being defined to be, that which has position but not magnitude. Here the affirmative part includes all that is essential to a point, and the negative part includes every thing that is not essential to it. I am indebted for this definition to a friend, by whose judicious and learned remarks I have often profited.

II.

After the second definition Euclid has introduced the following, "the "extremities of a line are points."

Now, this is certainly not a definition, but an inference from the definitions of a point and of a line. That which terminates a line can have no breadth, as the line in which it is has none; and it can have no length, as it would not then be a termination, but a part of that which is supposed to terminate. The termination of a line can therefore have no magnitude, and having necessarily position, it is a point. But as it is plain, that in all this we are drawing a consequence from two definitions already laid down, and not giving a new definition, I have taken the liberty of putting it down as a corollary to the second definition, and have added, that the intersections of one line with another are points, as this affords a good illustration of the nature of a point, and is an inference exactly of the same kind with the preceding. The same thing nearly has been done with the fourth definition, where that which Euclid gave as a separate definition is made a corollary to the

fourth, because it is in fact an inference deduced from comparing the definitions of a superficies and a line.

As it is impossible to explain the relation of a superficies, a line, and a point to one another, and to the solid in which they all originate, better than Dr. Simson has done, I shall here add, with very little change, the illustration given by that excellent Geometer.

"It is necessary to consider a solid, that is, a magnitude which has length, breadth, and thickness, in order to understand aright the definitions of a point, line and superficies; for these all arise from a solid, and exist in it; The boundary, or boundaries which contain a solid, are called superficies, or the boundary which is common to two solids which are contiguous, or which divides one solid into two contiguous parts, is called a superficies; Thus, if BCGF be one of the boundaries which contain the solid ABCDEFGH, or which is the common boundary of this solid, and the solid BKLCFNMG, and is therefore in the one as well as the other solid, it is called a superficies, and has no thickness; For if it have any, this thickness must either be a part of the thickness of the solid AG, or the solid BM, or a part of the thickness of each of them. It cannot be a part of the thickness of the solid BM; because, if this solid be removed from the solid AG, the superficies BCGF, the boundary of the solid AG, remains still the. same as it was. Nor can it be a part of the thickness of the solid AG: because if this be removed from the solid BM, the superficies BCGF, the boundary of the solid BM, does nevertheless remain; therefore the superficies BCGF has no thickness, but only length and breadth.

"The boundary of a superficies is called a line; or a line is the common boundary of two superficies that are contiguous, or it is that which divides one superficies into two contiguous parts: Thus, if BC be one of the boundaries which contain the superficies ABCD, or which is the common boundary of this superficies, and of the superficies KBCL, which is contiguous to it, this boundary BC is called a line, and has no breadth; For, if it have any, this must be part either of the breadth of the superficies ABCD or of the superficies KBCL, or part of each of them. It is not part of the breadth of the superficies KBCL; for if this superficies be removed from the superficies ABCD, the line BC which is the boundary of the superficies ABCD remains the same as it

was.

Nor can the breadth that BC is supposed to have, be a part of the breadth of the superficies ABCD; because, if this be removed from the superficies KBCL, the line BC, which is the boundary of the superficies

H

G

N

M

KBCL, does nevertheless remain: Therefore the line BC has no breadth. And because the line BC is in a superficies, and that a superficies has no thickness, as was shown; therefore a line has neither breadth nor thickness, but only length.

"The boundary of a line is called a point, or a point is a common boundary or extremity of two lines that are contiguous: Thus, if B be the ex