and accordingly we find that this porism is derived from the local theorem formerly given, by leaving out part of the hypothesis. But we shall afterwards have occasion to observe that this is an illogical and imperfect definition, not coextensive with the thing defined; the above proposition, however, answers every definition of a Porism.

The demonstration of the theorem or of the construction obtained by investigation in this manner of proceeding, is called synthesis, or composition, in opposition to the analysis, or the process of investigation; and it is frequently said that Plato imported the whole system in the visits which he made, like Thales of Miletus and Pythagoras, to study under the Egyptian geometers, and afterwards to converse with Theodorus at Cyrene, and the Pythagorean School in Italy. But it can hardly be supposed that all the preceding geometers had worked their problems and theorems at random; that Thales and Pythagoras with their disciples, a century and a half before Plato, and Hippocrates, half a century before his time, had no knowledge of the analytical method, and pursued no systematic plan in their researches, devoted as their age was to geometrical studies. Plato may have improved and further systematized the method, as he was no doubt deeply impressed with the paramount importance of geometry, and even inscribed upon the gates of the Lyceum a prohibition against any one entering who was ignorant of it. The same spirit of exaggeration which ascribes to him the analytical method, has also given rise to the notion that he was the discoverer of the Conic Sections; a notion which is without any truth and without the least probability.

Of the works written by the Greek geometers some have come down to us; some of the most valuable, as the Elements' and 'Data' of Euclid, and the 'Conics' of Apollonius. Others are lost; but, happily, Pappus, a mathematician of some merit, who flourished in the

Alexandrian school about the end of the fourth century, has left a valuable account of the geometrical writings of the elder Greeks. His work is of a miscellaneous nature, as its name, 'Mathematical Collections,' implies; and excepting a few passages, it has never been published in the original Greek. Commandini, of Urbino, made a translation of the whole six books then discovered; the first has never been found, but half the second being in the Savilian library at Oxford, was translated by Wallis a century later. Commandini's translation, with his learned commentary, was not printed before his death, but the Duke of Urbino (Francesco Maria) caused it to be published in 1588, at Pisa, and a second edition was published at Venice the next year: a fact most honourable to that learned and accomplished age, when we recollect how many years Newton's immortal work was published before it reached a second edition, and that in the seventeenth and eighteenth centuries.

The two first books of Pappus appear to have been purely arithmetical, so that their loss is little to be lamented. The eighth is on mechanics, and the other five are geometrical. The most interesting portion is the seventh; the introduction of which, addressed to his son as a guide of his geometrical studies, contains a full enumeration of the works written by the Greek geometers, and an account of the particular subjects which each treated, in some instances giving a summary of the propositions themselves with more or less obscurity, but always with great brevity. Among them was a work which excited great interest, and for a long time baffled the conjectures of mathematicians, Euclid's three books of 'Porisms:' of these we shall afterwards have occasion to speak more fully. 'Loci ad Superficiem,' apparently treating of curves of double curvature, is another, the loss of which was greatly lamented, the more because Pappus has given no account of its contents. This he had done in the

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case of the 'Loci Plani' of Apollonius. Euclid's four books on conic sections are also lost; but of Apollonius's eight books on the same subject, the most important of the whole series, the 'Elements' excepted, four were preserved, and three more were discovered in the seventeenth century. His Inclinations, his Tactions or Tangencies, his sections of Space and of Ratio, and his Determinate section, however curious, are of less importance; all of them are lost.

For many years Commandini's publication of the 'Collections' and his commentary did not lead to any attempt at restoring the lost works from the general account given by Pappus. Albert Girard, in 1634, informs us in a note to an edition to Stevinus, that he had restored Euclid's 'Porisms,' a thing eminently unlikely, as he never published any part of his restoration, and it was not found after his decease. In 1637, Fermat restored the 'Loci Plani' of Apollonius, but in a manner so little according to the ancient analysis, that we cannot be said to approach by means of his labours the lost book on this subject. In 1615, De la Hire, a lover and a successful cultivator of the ancient method, published his Conic Sections, but synthetically treated; he added afterwards other works on epicycloids and conchoids, treated on the analytical plan. L'Hôpital, at the end of the seventeenth century, published an excellent treatise on Conics, but purely algebraical. At the beginning of the eighteenth century, Viviani and Grandi applied themselves to the ancient geometry; and the former gave a conjectural restoration (Divinatio) of Aristaus's Loci Solidi,' the curves of the second or Conic order. But all these attempts were exceedingly unsuccessful, and the world was left in the dark, for the most part, on the highly interesting subject of the Greek geometry. We shall presently see that both Fermat and Halley, its most successful students, had made but an inconsiderable progress in the most difficult branches.

How entirely the academicians of France were either careless of those matters, or ignorant, or both, appears by the Encyclopédie'; the mathematical department of which was under no less a geometrician than D'Alembert. The definition there given of analysis makes it synonymous with algebra: and yet mention is made of the ancient writers on analysis, and of the introduction to the seventh book of Pappus, with only this remark, that those authors differ much from the modern analysts. But the article 'Arithmetic' (vol. i., p. 677) demonstrates this ignorance completely; and that Pappus's celebrated introduction had been referred to by one who never read it. We there find it said, that Plato is supposed to have invented the ancient analysis; that Euclid, Apollonius, and others, including Pappus himself, studied it, but that we are quite ignorant of what it was: only that it is by some conceived to have resembled our algebra, as else Archimedes could never have made his great geometrical discoveries. It is, certainly, quite incredible that such a name as D'Alembert's should be found affixed to this statement, which the mere reading of any one page of Pappus's books must have shown to be wholly erroneous; and our wonder is the greater, inasmuch as Simson's admirable restoration of Apollonius's Loci Plani' had been published five years before the 'Encyclopédie' appeared. Again, in the Encyclopédie,' the word Analysis, as meaning the Greek method, and not algebra, is not even to be found. Nor do the words synthesis, or composition, inclinations, tactions or tangencies, occur at all; and though Porisms are mentioned, it is only to show the same ignorance of the subject: for that word is said to be synonymous with 'lemma,' because it is sometimes used by Pappus in the sense of subsidiary proposition.* When Clairault wrote his inestimable work on curves of double curvature, he made no reference whatever to Euclid's 'Loci ad Superficiem,' * Euclid uses the word Corollary in his Elements.

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much less did he handle the subject after the same manner; he deals, indeed, with matters beyond the reach of the Greek geometry.

Such was the state of this science when Robert Simson first applied to it his genius, equally vigorous and undaunted, with the taste which he had early imbibed for the beauty, the simplicity, and the closeness of the ancient analysis.

ROBERT SIMSON was born on the 14th October (0.8.), 1687, at Kirton Hill, in the parish of Wester Kilbride, in Ayrshire. His father, John Simson, was a merchant in Glasgow; his grandfather, Patrick, was minister of Renfrew, and Dean of the Faculties in the University of Glasgow. Having been deprived at the Restoration, on being reinstated at the Revolution, he accompanied Principal Carstairs and a deputation as one of the Commissioners from the Kirk of Scotland to address the Sovereigns. Being a man of fine presence, it is related that the Queen and her maids of honour mistook him for the Principal, till the King set them right by presenting Carstairs to them. The grandson, Robert, is said to have been the eldest of seventeen children; and the estate of Kirton Hill, which had been in the family for several generations, being inconsiderable, it was necessary for him, as well as his brothers, to be placed in some profession. The assertion is made in one account, written by a son of Professor Millar, and is likely to be correct, that he was intended for the medical profession, and being sent to Leyden studied under Boerhaave. He appears to have been at first intended for the Church, and to have changed his plan. Dr. Traill, however, says, that he was always intended for the Church, and that when the University of St. Andrew's, in 1746, wished to confer on him a degree, they made him a Doctor of Medicine, because he had studied botany in his youth. Nothing can be more improbable than this story; for