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postulates are six in number (we translate literally from Euclid): 1. Let it be demanded from every point to every point to draw a straight line. 2. And to produce a terminated straight line continually in a straight line. 3. And with every centre and distance [from that centre) to draw a circle. 4. And that all right angles are equal to one another. 5. And that if a straight line falling on two straight lines make the angles within and towards the same parts less than two right angles, those two straight lines produced indefinitely will meet towards those parts at which are the angles less than two right angles. 6. And that two straight lines cannot enclose a space.

The common notions or opinions are: 1. Things equal to the same are equal to one another. 2. And if two equals be added the wholes are equal. 4. And if from equals equals be taken away, the remainders are equal. 4. And if to unequals equals be added, the wholes are unequal. 5. And if from unequals equals be taken away, the remainders are unequal. 6. And the doubles of the same are equal to one another. 7. And the halves of the same are equal to one another. 8. And things which fit one another are equal to one another. 9. And the whole is greater than the part.

The distinction drawn by Euclid, between that which the learner is now to grant, and the recapitulation of that which he always has granted, is clear and natural enough. Archimedes (in the sphere and cylinder) introduces, for the first time in geometry that we can find, the word axioms (ůžupara), things thought worthy (of something): the worthiness is worthiness to precede discussion, for the axioms of Archimedes are only definitions, pure verbal definitions, with mere statements preliminary to definition. Torelli translates the word pronuntiata, and Eutocius in his commentary fairly calls them definitions; his own postulates Archimedes calls haubavóueva, things taken. Geminus, according to Proclus, taking the distinction of theorem and problem, which was established by his time, though Euclid knew nothing about it (for mpútaois, proposition, is all the heading that Euclid gives), chose to fancy that a postulate and a common notion should become a postulate and an axiom; and that the postulate should be of the nature of a problem, something to be done; and an axiom of the nature of a theorem, something to be proved or made evident. Proclus wants to give into this idea, but had not enough of Robert Simson in him to alter his manuscript, in which five postulates existed, the sixth (two right lines cannot inclose a space) having been removed among the common notions by the writer. And thus Euclid rested, all (including the celebrated Vatican MSS.), except two, of the manuscripts of Peyrard;* some (he does not say how many) of those of Gregory; the Greek from which Zamberti took his Latin ; the printed Arabic; the translation of Adelard from the Arabic; the summary of Boethius, who suppresses the last postulate entirely; the newly-examined manuscripts of August;—place the fourth and fifth postulate as in the list given above, and many the sixth also. But Grynæus, for it cannot be traced higher, in the Basle edition, carried the views of Geminus into complete operation, and put the fourth and fifth postulates (as they were called) among common notions ! We do not know how far he was followed before the time of Gregory, not having thought it necessary to look over any more texts for the purpose of this article than those which give new readings; one only we have before us, the anonymous Greek of 1620, attributed to the celebrated Briggs, (Ward, p. 127) which follows Proclus, and gives five postulates. Gregory, who followed the Basle edition somewhat too often, coincided with Grynæus, against the practice of his predecessor Savile, who rather approved the notion of Geminus, but still allowed five postulates to remain. The texts of Peyrard and August have restored Euclid's six postulates, which seems to us common sense. Distinguish postulates into demanded problems and demanded theorems, if any one pleases, but in the name of arrangement, how can the celebrated demand in the theory of parallels rank under the same head as that “ things which are equal to the same are equal to one another.” The misplacement of this axiom about parallels has cost many a trial at this old difficulty, and procured Euclid all manner of reproaches which he did not deserve. He has been made to say, "I give you this common notion as a most self-evident theorem;" whereas he only said, “whether this be easy to you or not, I can't proceed till you grant it.” And let it be observed, that none of the opponents of Euclid's text cast a thought upon the absence of “axioms," and the use of “common notions." The word axiom had got into their heads : thus Barrow, after a long and cloudy lecture about principles,

* In nine manuscripts (the Vatican included) the fourth and fifth are postulates; in none, common notions. In four manuscripts (the Vatican included) the sixth is a postulate; in seven, a common notion.

axioms, &c. with a full consideration of Aristotle, Proclus, &c. decides that Euclid was inaccurate (hinting at the same time a doubt of the correctness of the text) when he made a simple demand, and called it a demand.

Such is a specimen of the manner in which the text of Euclid has been handled, and it will make many persons doubt whether they have ever read that writer, with whom till now they have supposed themselves well acquainted. We can assure them, however, that Robert Simson is, when he translates, as good a translator as he might have been a critic, if he had not had that unfortunate dream about Theon which we have related. He, or any editor, might judiciously have practised something like condensation after the first book; for from first to last, Euclid fights every step of the way as if he were arguing with an opponent who would never see one iota more than he was obliged to do. And in all probability this was actually the case. Watch Proclus's account narrowly, and it will appear most probable that this work of Euclid ushered connected demonstration into the world. We

may

think it very likely then that the prominent idea before Euclid's mind was, not “ this proposition can be demonstrated,” but “ there is such a thing as demonstration." To such a leading notion it would matter nothing what the definitions were, as long as they were well understood between the two parties; nor what the postulates were, as long as they were what no one of the time objected to. Neither would it matter that every postulate should be expressed, since, in the absence of any thing like previous guide, it would be natural to insist only on those preliminaries which had already been agitated in the previous attempts which we must imagine to have been made. It is only in some such way that we can give anything like a surmise at the reason why Euclid has really several more postulates than the six which he places at the beginning of his work. For example, that if of two bounded figures, one be partly inside and partly outside the other, the boundaries must somewhere intersect, is a very admissible postulate, but quite as necessary to be mentioned as that two straight lines cannot inclose a space. This is taken for granted without mention in the very first proposition. Again, that if two straight lines meet in a point, they will if produced cut in that point; that a straight line of which any one point is within a bounded figure, must, if produced indefinitely, cut that figure in two points; that if two points lying on opposite sides of a straight line be joined, the joining line must cut the straight line; that two circles may coincide in one point only, one of them being entirely within, or entirely without, the other; and perhaps some others—are all tacitly assumed. As to common notions, we might instance “ things which areunequal to one another cannot be equal to the same, which is frequently used, and might be set down in a list which contains “ the whole is greater than its part.” It is not easy to see any probable reason for Euclid's preliminary selection, unless it be admitted as such, that he was writing on the point of demonstration generally, with reference to some particular opponents, whose requisitions he knew, or thought he knew.

All the earlier editions of Euclid announce him to be Euclid of Megara, who founded a sect of philosophers in that town. Diogenes Laertius, Suidas, and Aulus Gellius, give some account of Euclid of Megara, but not as a geometer: Proclus and Heron, who give an account of the geometer, do not mention Megara : Plutarch alone calls Euclid of Megara a geometer. It

may therefore be concluded that the verdict of later times is correct; and that the philosopher of Megara is altogether a distinct

We must now conclude an article which the bibliographer may think too concise, and the general reader too long. What do people care about old books and old editions ? Little enough we are obliged to admit,-as little, in fact, as they care about accurate history. But every now and then an historical article is bearable; and many persons may just feel that degree of interest in Euclid which will enable them to glance at an account of the writer about whom they doubted when they were boys, whether his name was that of a science or of a man. Let them doubt on this point still, as much as they please, on condition that there shall be no coalition of the two designations, no joining of the names. May all good powers protect us from ever hearing Euclid called a man of science ! We once read of him in a French book as ce savant distingué, and must confess we did not feel in a concatenation accordingly. But to return to old books: there are about them indications of old times which may be worthy subjects of ridicule to the modern man, who will himself be looked at in a similar light when his time shall come; or rather when his time shall be past, and the time of others shall come. What will our speechifiers at public meetings say to one which was held on the eleventh of August, 1508, in the church of St. Bartholomew at Venice ;--present, the Rev.

person.

Lucas Pacioli,* of the order of minorite Franciscans, in the chair; the diplomatic ministers of France and Spain ; various men of learning not otherwise distinguishable; seventeen ecclesiastical functionaries; ten doctors and professors; fiftynine physicians, poets, printers, (including the celebrated Aldus), and gentlemen without title; besides citizens of Venice. The meeting being constituted, the reverend chairman proceeded to business, namely, the opening of his explanations of the fifth book of Euclid. His address (of which we regret we have not room for a full report) was with some few exceptions (among which we may number his statements of the necessity of the doctrines of proportion to a full understanding of those of religion) as much to the purpose as if it had been delivered immediately after dinner at the London Tavern, or at any period of the day at Exeter Hall: at least after making due allowance for his profession, which prevented him from speaking against the Catholics, and for his utter ignorance of Irish affairs. The effect of his explanation was to induce one of the ecclesiastics present to declare by letter to another, that the fifth book of Euclid excelled ail the others as much as those others excelled the writings of other men.

This we know, because, oddly enough, the account of this public meeting, with the names of the persons present, and the letter just alluded to annexed (dated March 12, 1509), is inserted bodily in the edition of Euclid published (or at least finished) by Fra Lucas himself, June 21, 1509. It sticks between the fourth and fifth books; and looking at the date of the letter and that of the completion of the work, it appears that two hundred and thirty folio pages of close black letter were composed, or at least revised, in less than half the number of days. Oh Lucas Pacioli! what would he have said if he could have known that his lectures would have been one day dragged from their obscurity to prove nothing but the rate at which printing went on in his day.

* This gentleman, under the name of Lucas di Borgo, is a personage in the history of algebra ; but those who persist in calling him Di Borgo, might just as well call Hobbes by the appellation of “ Hobbes of,” leaving out “ Malmesbury.” Lucas Paciolus de burgo Sancti Sepulcri, is his proper title.

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