which follow, appears sufficiently from a consideration which was long ago mentioned by Locke,-that from those axioms it is not possible for human ingenuity to deduce a single inference. "It was not (says Locke) the influence of those maxims which are taken for principles in mathematics that hath led the masters of that science into those wonderful discoveries they have made. Let a man of good parts know all the maxims generally made use of in mathematics never so perfectly, and contemplate their extent and consequences as much as he pleases, he will, by their assistance, I suppose, scarce ever come to know, that the square of the bypothenuse in a right angled triangle, is equal to the squares of the two other sides.' The knowledge that the whole is equal to all its parts,' and, if you take equals from equals, the remainders will be equal,' helped him not, I presume, to this demonstration: And a man may, I think, pore long enough on those axioms, without ever seeing one jot the more of mathematical truths."* But surely, if this be granted, and if, at the same time, by the first principles of a science be meant, those fundamental propositions from which its remoter truths are derived, the axioms cannot, with any consistency, be called the First Principles of Mathematics. They have not (it will be admitted) the most distant analogy to what are called the first principles of natural philosophy;—to those general facts, for example, of the gravity and elasticity of the air, from which may be deduced, as consequences, the suspension of the mercury in the Torricellian tube, and its fall when carried up to an eminence. According to this meaning of the word, the principles of mathematical science are, not

* Essay on Human Understanding, Book IV. chap. xii. § 15.

the axioms but the definitions; which definitions hold, in mathematics, precisely the same place that is held in natural philosophy by such general facts as have now been referred to.*

From what principle are the various properties of the circle derived, but from the definition of a circle? From what principle the properties of the parabola or ellipse, but from the definitions of these curves? A similar observation may be extended to all the other theorems which the mathematician demonstrates: And it is this observation (which, obvious as it may seem, does not appear to have

In order to prevent cavil, it may be necessary for me to remark here, that when I speak of mathematical axioms, I have in view only such as are of the same description with the first nine of those which are prefixed to the Elements of Euclid; for, in that list, it is well known, that there are several which belong to a class of propositions altogether different from the others. That "all right angles (for example) are equal to one another;" that "when one straight line falling on two other straight lines makes the two interior angles on the same side less than two right angles, these two straight lines, if produced, shall meet on the side, where are the two angles less than two right angles ;" are manifestly principles which bear no analogy to such barren truisms as these, "Things that are equal to one and the same thing are equal to one another." "If equals be added to equals, the wholes are equal." "If equals be taken from equals, the remainders are equal." Of these propositions, the two former (the 10th and 11th axioms, to wit, in Euclid's list) are evidently theorems which, in point of strict logical accuracy, ought to be demonstrated; as may be easily done, with respect to the first, in a single sentence. That the second has not yet been proved in a simple and satisfactory manner, has been long considered as a sort of reproach to mathematicians; and I have little doubt that this reproach will continue to exist, till the basis of the science be somewhat enlarged, by the introduction of one or two new definitions, to serve as additional principles of geometrical reasoning.

For some farther remarks on Euclid's Axioms, see note (A.)

The edition of Euclid to which I uniformly refer, is that of David Gre gory. Oxon. 1713.

occurred, in all its force, either to Locke, to Reid, or to Campbell,) that furnishes, if I mistake not, the true explanation of the peculiarity already remarked in mathematical evidence.*

The prosecution of this last idea properly belongs to the subject of mathematical demonstration, of which I intend to treat afterwards. In the meantime, I trust, that enough has been said to correct those misapprehensions of the nature of axioms, which are countenanced by the speculations, and still more by the phraseology, of some late eminent writers. On this article, my own opinion coincides very nearly with that of Mr. Locke-both in the view which he has given of the nature and use of axioms in geometry, and in what he has so forcibly urged concerning the danger, in other branches of knowledge, of attempting a similar list of maxims, without a due regard to the circumstances by which different sciences are distinguished from one another. With Mr. Locke, too, I must beg leave to guard myself against the possibility of being misunderstood in the illustrations which I have offered of some of his ideas: And for this purpose, I cannot do better than borrow his words. "In all that is here sug

* D'Alembert, although he sometimes seems to speak a different language, approached nearly to this view of the subject when he wrote the following passage:

"Finally, it is not without reason that mathematicians consider definitions as principles; since it is on clear and precise definitions that our knowledge rests in those sciences, where our reasoning powers have the widest field opened for their exercise."-" Au reste, ce n'est pas sans raison que les mathématiciens regardent les définitions comme des principes, puisque, dans les sciences ou le raisonnement a la meilleure part, c'est sur des définitions nettes et exactes que nos connoissances sont appuyées."-Elemens de Phil. p. 4.

gested concerning the little use of axioms for the improvement of knowledge, or dangerous use in undetermined ideas, I have been far enough from saying or intending they should be laid aside, as some have been too forward to charge me. I affirm them to be truths, selfevident truths; and so cannot be laid aside. As far as their influence will reach, it is in vain to endeavour, nor would I attempt to abridge it. But yet, without any injury to truth or knowledge, I may have reason to think their use is not answerable to the great stress which seems to be laid on them, and I may warn men not to make an ill use of them, for the confirming themselves in error."*

After what has been just stated, it is scarcely necessary for me again to repeat, with regard to mathematical axioms; that although they are not the principles of our reasoning, either in arithmetic or geometry, their truth is supposed or implied in all our reasonings in both; and, if it were called in question, our further progress would be impossible. In both of these respects, we shall find them analogous to the other classes of primary or elemental truths which remain to be considered.

Nor let it be imagined, from this concession, that the dispute turns merely on the meaning annexed to the word principle. It turns upon an important question of fact; Whether the theorems of geometry rest on the axioms, in the same sense in which they rest on the definitions? or (to state the question in a manner still more obvious,) Whether axioms hold a place in geometry at

* Locke's Essay, Book IV. ch. vii. § 14.

all analogous to what is occupied in natural philosophy, by those sensible phenomena which form the basis of that science? Dr. Reid compares them sometimes to the one set of propositions and sometimes to the other.* If the foregoing observations be just, they bear no analogy to either.

Into this indistinctness of language Dr. Reid was probably led in part by Sir Isaac Newton, who, with a very illogical latitude in the use of words, gave the name of axioms to the laws of motion,† and also to those general ex

* "Mathematics, once fairly established on the foundation of a few axioms and definitions, as upon a rock, has grown from age to age, so as to become the loftiest and the most solid fabric that human reason can boast.”Essays on Int. Powers, p. 561, 4to edition.

"Lord Bacon first delineated the only solid foundation on which natural philosophy can be built; and Sir Isaac Newton reduced the principles laid down by Bacon into three or four axioms, which he calls regulæ philosophandi. From these, together with the phenomena observed by the senses, which he likewise lays down as first principles, he deduces, by strict reasoning, the propositions contained in the third book of his Principia, and in his Optics; and by this means has raised a fabric, which is not liable to be shaken by doubtful disputation, but stands immoveable on the basis of self-evident principles."-Ibid. See also pp. 647, 648.

† Axiomata, sive leges Motus. Vid. Philosophia Naturalis Principia Mathematica.

At the beginning too, of Newton's Optics, the title of axioms is given to the following propositions:

AXIOM I.

"The angles of reflection and refraction lie in one and the same plane with the angle of incidence.

AXIOM II.

"The angle of reflection is equal to the angle of incidence.

AXIOM III.

"If the refracted ray be turned directly back to the point of incidence, it shall be refracted into the line before described by the incident ray.