domestic government of the state, where the public good, which is, or ought to be, their object, depends on the concurrence of a multitude of causes; not, as in foreign politics, on accidents and chances, and the caprices of a few persons."

"1

To these profound reflections of Mr. Hume, it may be added, (although the remark does not bear directly on our present argument,) that, in the systematical application of general and refined rules to their private concerns, men frequently err from calculating their measures upon a scale disproportionate to the ordinary duration of human life. This is one of the many mistakes into which projectors are apt to fall; and hence the ruin which so often overtakes them, while sowing the seeds of a harvest which others are to reap. A few years more might have secured to themselves the prize which they had in view, and changed the opinion of the world (which is always regulated by the accidental circumstances of failure or of success) from contempt of their folly, into admiration of their sagacity and perseverance.

It is observed by the Comte de Bussi, [Bussy Rabutin ?] that "time remedies all mischances; and that men die unfortunate, only because they did not live long enough. Mareschal d'Estrées, who died rich at a hundred, would have died a beggar, had he lived only to eighty." The maxim, like most other apothegms, is stated in terms much too unqualified; but it may furnish matter for many interesting reflections to those who have surveyed with attention the characters which have passed before them on the stage of life; or who amuse themselves with marking the trifling and fortuitous circumstances by which the multitude are decided, in pronouncing their verdicts of foresight or of improvidence.

1 Essay on Commerce.

This contrast between the domestic and the foreign policy of a state occurs more than once in Mr. Hume's writings. (See in particular the first paragraphs of his Essay on the Rise of Arts and Sciences.) A similar observation had long before been made by Polybius.

"There are two ways by which every kind of government is destroyed; either by some accident that happens from without, or some evil that arises within itself: When the first will be, it is not always easy to foresee: but the latter is certain and determinate."-Book vi. Ex. 3. (Hampton's Translation.)

SUBSECTION] IV.-Continuation of the Subject.-Peculiar and supereminent Advantages possessed by Mathematicians, in consequence of their definite Phraseology.

If the remarks contained in the foregoing articles of this section be just, it will follow, that the various artificial aids to our reasoning powers which have been projected by Leibnitz and others, proceed on the supposition (a supposition which is also tacitly assumed in the syllogistic theory) that, in all the sciences, the words which we employ have, in the course of our previous studies, been brought to a sense as unequivocal as the phraseology of mathematicians. They proceed on the supposition, therefore, that by far the most difficult part of the logical problem has been already solved. Should the period ever arrive, when the language of moralists and politicians shall be rendered as perfect as that of geometers and algebraists, then, indeed, may such contrivances as the Ars Combinatoria and the Alphabet of human thoughts, become interesting subjects of philosophical discussion; although the probability is, that, even were that era to take place, they would be found nearly as useless, in morals and politics, as the syllogistic art is acknowledged to be at present in the investigations of pure geometry.

Of the peculiar and supereminent advantage possessed by mathematicians, in consequence of those fixed and definite relations which form the objects of their science, and the correspondent precision in their language and reasonings, I can think of no illustration more striking than what is afforded by Dr. Halley's Latin version from an Arabic manuscript, of the two books of Apollonius Pergæus, De Sectione Rationis. The extraordinary circumstances under which this version was attempted and completed, (which I presume are little known beyond the narrow circle of mathematical readers,) appear to me so highly curious, considered as matter of literary history, that I shall copy a short detail of them from Halley's preface.

After mentioning the accidental discovery in the Bodleian Library, by Dr. Bernard, Savilian Professor of Astronomy, of

the Arabic version of Apollonius, Περὶ Λόγου ̓Αποτομῆς, Dr. Halley proceeds thus:

"Delighted, therefore, with the discovery of such a treasure, Bernard applied himself diligently to the task of a Latin translation. But before he had finished a tenth part of his undertaking, he abandoned it altogether, either from his experience of its growing difficulties, or from the pressure of other avocations. Afterwards, when on the death of Dr. Wallis, the Savilian professorship was bestowed on me, I was seized with a strong desire of making a trial to complete what Bernard had begun ;-an attempt, of the boldness of which the reader may judge, when he is informed, that in addition to my own entire ignorance of the Arabic language, I had to contend with the obscurities occasioned by innumerable passages which were either defaced or altogether obliterated. With the assistance, however, of the sheets which Bernard had left, and which served me as a key for investigating the sense of the original, I began first with making a list of those words, the signification of which his version had clearly ascertained; and then proceeded, by comparing these words wherever they occurred, with the train of reasoning in which they were involved, to decypher by slow degrees the import of the context, till at last I succeeded in mastering the whole work, and in bringing my translation (without the aid of any other person) to the form in which I now give it to the public."1

When a similar attempt shall be made, with equal success, in decyphering a moral or a political treatise, written in an unknown tongue, then, and not till then, may we think of comparing the phraseology of these two sciences with the simple and rigorous language of the Greek geometers, or with the more refined and abstract, but not less scrupulously logical system of signs, employed by modern mathematicians.

It must not, however, be imagined, that it is solely by the nature of ideas which form the objects of its reasonings, even when combined with the precision and unambiguity of its

1 Apollonius Pergæus, De Sectione Rationis, &c. Opera et Studio Edmundi Halley. Oxon. 1706. In Præfat.

phraseology, that mathematics is distinguished from the other branches of our knowledge. The truths about which it is conversant, are of an order altogether peculiar and singular; and the evidence of which they admit resembles nothing, either in degree or in kind, to which the same name is given, in any of our other intellectual pursuits. On these points also, Leibnitz and many other great men have adopted very incorrect opinions; and by the authority of their names, have given currency to some logical errors of fundamental importance. My reasons for so thinking I shall state, as clearly and fully as I can, in the following section.

SECTION III.-OF MATHEMATICAL DEMONSTRATION.

[SUBSECTION] I.-Of the Circumstance on which Demonstrative
Evidence essentially depends.

The peculiarity of that species of evidence which is called demonstrative, and which so remarkably distinguishes our mathematical conclusions from those to which we are led in other branches of science, is a fact which must have arrested the attention of every person who possesses the slightest acquaintance with the elements of geometry. And yet I am doubtful if a satisfactory account has been hitherto given of the circumstances from which it arises. Mr. Locke tells us, that "what constitutes a demonstration is intuitive evidence at every step;" and I readily grant, that if, in a single step, such evidence should fail, the other parts of the demonstration would be of no value. It does not, however, seem to me that it is on this consideration that the demonstrative evidence of the conclusion depends-not even when we add to it another which is much insisted on by Dr. Reid-that, "in demonstrative evidence our first principles must be intuitively certain." The inaccuracy of this remark I formerly pointed out when treating of the evidence of axioms,* on which occasion I also observed, that the first principles of our reasonings in mathematics are not axioms, but definitions. It is in this last circumstance (I mean the *P. 32, before and after.-Ed.

VOL. III.

H

peculiarity of reasoning from definitions) that the true theory of mathematical demonstration is to be found, and I shall accordingly endeavour to explain it at considerable length, and to state some of the more important consequences to which it leads. That I may not, however, have the appearance of claiming in behalf of the following discussion, an undue share of originality, it is necessary for me to remark, that the leading idea which it contains has been repeatedly started, and even to a certain length prosecuted by different writers, ancient as well as modern; but that, in all of them, it has been so blended with collateral considerations, altogether foreign to the point in question, as to divert the attention both of writer and reader, from that single principle on which the solution of the problem hinges. The advantages which mathematics derives from the peculiar nature of those relations about which it is conversant, from its simple and definite phraseology, and from the severe logic so admirably displayed in the concatenation of its innumerable theorems, are indeed immense, and well entitled to a separate and ample illustration, but they do not appear to have any necessary connexion with the subject of this section. How far I am right in this opinion, my readers will be enabled to judge by the sequel.

It was already remarked, in the first chapter of this part, that whereas, in all other sciences, the propositions which we attempt to establish express facts real or supposed—in mathematics, the propositions which we demonstrate only assert a connexion between certain suppositions and certain consequences. Our reasonings, therefore, in mathematics, are directed to an object essentially different from what we have in view, in any other employment of our intellectual faculties,—not to ascertain truths with respect to actual existences, but to trace the logical filiation of consequences which follow from an assumed hypothesis. If from this hypothesis we reason with correctness, nothing, it is manifest, can be wanting to complete the evidence of the result; as this result only asserts a necessary connexion between the supposition and the conclusion. In the other sciences, admitting that every ambiguity of language were