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to this absurd tale. There are some who have thought that they had got a glimpse of it from an experiment which they took a great deal of pains about at the end of the last age. I mean the transfusion of blood, a remedy which they tried many times with ill success. Others search for the origin of that fable in a tradition which imports, that Medea knew herbs, whose virtue was to make white hairs black. But all these explications are not supported on any historical foundations p.
THE *HE Greeks, in the ages of which we at present speak,
had only very contracted notions of mathematics. What they knew in it does not merit the name of science. We are always astonished, when we compare the brilliant ages of that nation with its beginnings. Their genius has been far from being unfolded as readily as that of the people of the east. Compare the Greeks of the heroic ages to the Phoenicians of the same ages, and we shall find almost as much difference between them as between the most policed people of Europe, and the nations of America the moment they 'were discovered. The Greeks even did not know to put in practice, till very lately, the knowledge which the Asian and Egyptian colonies had imparted to them. However imperfect we suppose these first tinctures, the little use which the Greeks made of them for almost 1000 years will always be a great subject of astonishment.
Bannier explic. des fables, t. 6. p. 459. & 460. • Clem. Alex. ftrom. 1. 1. p. 363. See le Clerc hist, de la medecine, 1. 1. p. 65.
P Bannier, loco cit, p. 460.
It is impossible to give even imperfe& and vague notions
of the state and progress of arithmetic in Greece for the heroic ages. Antiquity does not furnish us with any lights about the first methods that the Greeks had made use of to make their calculations. I shall content myself with proposing some conjectures about the arithmetical symbols used anciently among these people.
The Greeks, like all the nations of antiquity, had no knowledge of figures properly so called, that is to say, characters solely deitined to express numbers. They made serve for this purpose the letters of their alphabet, divided and ranged in different manners. - It appears, that at first they designed numbers by the initial letters *, to which they afterwards added the numeral letters 4. The first being, if one may say so, only the abridgment of the names of number, they ought to have made use of them before they gave to the letters of the alphabet a value dependent, not only of the rank which they held, but even an arbitrary agreement, which is plain from the manner of expressing units, tens, hundreds, &c. This second operation is much more complicated than the first. It could not be introduced, till they had received from the Phoenicians
* This method could not have had place in the case where the same initial letter agreed to many names of different numbers. It would be difficult, for example, to make use of Epsilon, for the numbers fix, Seven, nine, 's, , évvéce, when it was necessary to express them in one and the same calculation. They must necessarily, in that cale, have had error and confusion, to design those numbers by the initial letter of their name. We are ignorant in what manner the Greeks in the first ages remedied this inconveniency. But the monuments which still subfift, do not permit us to doubt of the great use they made, generally speaking, of initial letters, of the names of numbers to express their value in an abridged way. • See les mem, de l'acad. Jes inscript. t. 23. mem. p. 496. Ga
the Epifemons, Bai, Koppa, and Sampi *, which appear to have come later into Greece than the greatest part of the other characters.
In the times of Herodian, the first manner of reckoning still existed in the laws of Solon, and on ancient columns'. It was perpetuated among the Athenians ; but, as it had been insensibly abandoned by the other cities of Greece, from thence it comes, that the grammarians, such as Terentius Scaurus, and Priscian; never speak of it but as a custom particular to the Athenians s.
It is clear, notwithstanding, that, at the beginning, this custom must have been common to all the people of Greece.' We find proofs of it in some fragments of very ancient inscriptions But we must agree at the same time, that the other method of reckoning, that is to say, by numeral letters, was introduced very early into many districts of Greece u.
I should like to have been able to have spoken more ex.
* It is the name which the Greeks gave to three characters, which they added to the 24 letters of their alphabet, to extend and facilitate the practice of calculations. These characters were formed thus s, 5, 7), and repreTented the numbers 6, 90, & 900. The 24 letters of the alphabet, taken according to the order that they had given to them originally, marked the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 85, 100, 200, 300, 400, 500, 600, 700, & 800. The combination of the cight letters á, x,a's pé', ', 5, 6, 7t', and of Koppa , with the first eight cá, b, 7, 8, 6, 6's “, d', and with the episemon Bau, served to express all the intermediate numbers between 10 & 25, between 20 & 30, and thus following to an hundred. Lastly, the eight letters
6, 5, 6', 6, 9":%, fà, and the Sampi ), combined together as well with the six preceding and the two first épisemons, as with the combinations of the first eight augmented with Bau, and with the eight intermediate ones, augmented with Koppa, express all the numbers which are between 100 & 200, between 200 & 300, &c. to 1000. All these characters, as well simple as compound, were accented at the top.
To express all the numbers which are between 1000 & 1000000, they did not use new numerical symbols, they contented themselves with only remo. ving the accent to the interior part of the character, which without that on. ly meant units, tens, hundreds; this new position of the accent determined the character to represent units, tens, and hundreds of thousands.
See his treatife περί των αριθμών.
Terent. Scaurus de orth. p. 2258. edit. de Putr. ; Priscus, de fig. num.
Ibid. loco cit.
tenlively of the origin and state of arithmetic among the Greeks in these early ages. The silence of ancient authors has not permitted me. It would be difficult to supply it by conjectures, which besides would necessarily have this defect, to be very uncertain and very arbitrary. Aftronomy will furnish us with more matter for our researches.
Nothing fliews better the little disposition the ancient
Greeks had for the sciences, than the state of im. perfection in which astronomy had languished among them during so many ages. It is certain, that at the times of which we now speak, and very long after them, their calendar was very imperfect. It was, without doubt, be. cause the Greeks did not give themselves up to agriculture till pretty late, and that they had been a very long time without undertaking navigations of a great extent z.
It appears nevertheless, that that nation had never wanted astronomers. The greatest part of the famous perfonages of the heroic ages were said to have applied them. selves to the study of the heavens. There is scarce one of them, to whom they have not attributed some astronomical discoveries y. If we would even believe Philostrates, Palamedes had been instructed enough in that science to explain the cause of the eclipses of the sun 2, I have al. ready sufficiently explained myself as to what we thould think of the pretended discoveries of these heroes; it would then be losing of time to stop any longer about it.
There is great reason to think, that, in the beginning, the Greeks did not reckon their years but by the seasons; and yet there was not, in that respect, a uniformity between
* See supra, book 2. chap. 1. p. 174. &c. et infra book 4. chap. 4. > See Lucian, de astrol. t. 2. p. 361. &feq.; Achil. Tat. Ifag. init.
Heroic. c. 10. p. 709.
the different people of Greece. The Arcadians, who pala fed for the first who had endeavoured to make themselves a calendar, originally made the year of three months, and afterwards of four. The Argives and the Acarnanians gave six to theirs.
We cannot fix the age in which the Greeks came to accommodate in a reasonable way the duration of their years to the course of the seasons. Anciently their years were purely lunar 6. The Greeks were not long of perceiving how irregular that manner of dividing the time was. In less than seventeen of these years, the order of nature was abfolutely reversed; summer taking the place of winter, and winter that of fummer. They were obliged to have a remedy for these inconveniencies. The Greeks invented fucceffively different periods, or cycles, to make the dura. tion of their years concur with the periodical return of the seasons ; but they wanted the most essential sciences, without which it was not poffible to succeed in such an enterprise. We have a striking proof of this, even in the nature of these periods. The first was the Dieteride.
This period supposed that twenty-five lunar revolutions answered exactly to two solar revolutions. In consequence of this false principle, the Greeks believed they had found the true means of bringing back again the different months of their year to the fame feafon, by interealating a thirteenth month every other two years, in such a way that the years were alternatively of twelve and of thirteen months <. They called that period Dieteride or Trieteride, that is to say, a period of two years, or a period of three years, because that intercalation did not take place but each third year, after two years revolution
a Plin. 1.7. c. 48. p. 403; ; Censorin. c. 19.; Solin, c. 1. p. 4.; Plut. in Numa, p. 72. B.; Stob.eclog. phyl.p. 21.; Auguft. de civit. Dei, l. 15. C. 12. p. 129. ; Macrob. Saturn. 1. 1. C. 12. p. 242.
• Solin, c. 1. p. 4. ; Suid in 'Evlavròs, t. 1. p. 747.; Macrov, Saturn. l. 1. c. 12. p. 242. C. 13. p. 251.
We shall afterwards fee the proof of what we are going to report of their ancient periods, which necessarily suppose lunar years of 354 days. ç Censorin. c. 18.