to 
NATURE 
| May 7, 1885 
viz. Bulletin des Sciences Math. et Astronomigues, is an 
easily accessible one, and we think from the analyses we 
have from time to time given in these columns of other 
papers by M. Tannery (in Mémotres de la Société des 
Sciences Physigues et Naturelles de Bordeaux) on Greek 
arithmetic and geometry, our author would have gathered 
useful material in the Bz//etin paper and in the A/émozres 
also.!_ But this is the only case of omission we have come 
across; the reading is apparently most thorough, and 
the author’s Greek scholarship enables him to improve 
upon the translations given by some of these foreign 
authorities. 
The work consists of three parts. The first part, 
entitled “ Prolegomena to Arithmetic,’ discusses the 
decimal scale and Egyptian arithmetic in a very thorough 
manner. Here, of course, much use is made of the 
“Rhind papyrus,” a book written by one Ahmes (now put 
at 1700 B.C.), entitled “Directions for Obtaining the 
Knowledge of all Dark Things,” which consists mainly of 
statements of results. One could wish some safe means 
could be discovered by the Museum authorities for un- 
folding the “olden leather-roll on a mathematical sub- 
ject ” which is “apparently too stiff to be opened ” ! (note, 
p- 16). 
There are naturally statements in these early chapters 
which are fairly open to objection, but they are clearly 
put, and the results, as Mr. Gow gets them, are sum- 
marised on pp 20, 21. 
The second part treats of Greek arithmetic under “‘ Lo- 
gistica, or Calculation and Arithmetica, or Greek Theory 
of Numbers.” This part is very carefully done, and 
enables the reader to get a clear idea of the processes 
employed. Plato’s appreciation of /ogistée may be in- 
ferred from his direction (Legg, 819 8) that “free boys 
shall be taught calculation, a purely childish art, by 
pleasant sports, with apples, garlands, &c.”* 
The third part treats of Greek geometry, and upon it 
we could expatiate at some length, but that is hardly our 
business on the present occasion. We need only say that 
there is much good work. Dr. Allman’s powerful rectifi- 
cation of the position of Eudoxus did not appear in time 
to be of service to Mr. Gow (he mentions the fact of its 
publication on p. x. of the Acdenda). Most of the geo- 
meters appear to have justice done them. We miss some 
of the touches which appear in M. Marie’s work, but 
again we find a compensation in the fuller account given 
of Menelaus, and of the proposition now usually cited 
by the name of that geometer. Chapter V. discusses 
“prehistoric and Egyptian geometry,” in which is given 
an account of Ahmes’ work. Chapter VI. takes “ Greek 
Geometry to Euclid” in five sections. Of the Pytha- 
gcreans, the Eudemian summary (which has in previous 
numbers been referred to in our notices of Dr. Allman’s 
papers) says they made geometry “a liberal education ;” 
and other writers, referred to by Mr. Gow, attribute to 
them the maxim, “A figure and a stride: not a figure 
and sixpence gained” (p. 153). In connection with this 
characteristic maxim we may give the story, which, in the 
Greek, forms the motto on the title-page of Mr. Gow’s 
* In the Buéletin for March, 1985, there is a paper by M. Tannery, ‘ Sur 
l’Arithmétique Pythagorienne ” (pp. 69 88). 
* It is curious to note that there was a Cocker before Cocker: cf. the 
Lucianic compliment, ‘‘You reckon like Nicomachus of Gerasa”; see also, 
in the opposite direction, ‘‘ Budget of Paradoxes,” p. 30. 
book, viz. “A youth who had begun to read geometry 
with Euclid, when he had learnt the first proposition, 
inquired ‘What do I get by learning these things?’ So 
Euclid called his slave, and said: ‘ Give him threepence, 
since he must make a gain out of what he learns.’” Many 
such boys there are, even in this nineteenth century, who 
are ever asking, “ What is the use of learning Euclid?” 
We thank Mr. Gow for his story from Stobzeus, which 
will possibly make us better prepared to answer the 
question the next time we are asked it. There is much 
other quotable matter, but we hasten to a close. Chap- 
ter VII. gives an account of Euclid (what little is known. 
of him, his writings, history of text of “ Elements,” and 
modern history of the book®), Archimedes, and Apollo- 
nius. Chapter VIII. is on “Geometry in Second Cent- 
ury B.C. ;” Chapter IX., “ From Geminus to Ptolemy ; ” 
and Chapter X., “Lost Years,” principally occupied 
with an account of Pappus and his “ Mathematice 
Collectiones.” 
Some matters of interest ave illustrated, as the intro- 
duction of the szgvzs in algebra, of the séze in trigono- 
metry (it does not seem to be generally known that the 
first occurrence of “tangent” and “ secant” is traced by 
De Morgan toa work by T. Finkius, “ Geometriz rotundi 
libri xiiii.,” Basileaze, 1583), the derivation of “ almagest ” 
(cf. Chaucer’s Clerk Nicholas, who had— 
‘* His almageste and bokes grete and s nall, 
His astrelabre, longing for his art, 
His augrim stones, layen faire apart 
On shelves couched at his beddes head”) 
and a few others. 
On page 290, line 9 up, for An read a». 
OUR BOOK SHELF 
The Zoological Record for 1833, being Volume XX. of 
the Record of Zoological Literature. "Edited by E. C. 
Rye, F.Z.S., &c. (London: John Van Voorst. 1884.) 
ALTHOUGH bearing on its title-page the date 1884, it was 
not until the end of January in this year that the “ Zoolo- 
gical Record for 1883 ” was, in its entire form, laid before 
the public. It comes to us with a melancholy interest, as 
being the last under the editorship of the late Mr. Rye, 
whose untimely death we have so recently recorded and 
deplored. Again in this volume we have to mention still 
further changes in the staff of the Recorders. Prof. 
Sollas takes Mr. S. O. Ridley’s place as recording the 
sponges, and Prof. Haddon that of Mr. W. Saville Kent 
in recording the Protozoa. Other engagements have 
prevented the Rev. O. P. Cambridge recording the litera. 
ture of the Arachnids for 1883, and it has been arranged 
that Mr. T. D. Gibson-Carmichael is to record the litera- 
ture of this group for 1883 and 1884 in the next volume 
of the “ Record.” 
A rapid glance over the contents of the volume brings 
to light the fact that in all the leading groups of the ani- 
mal kingdom a goodly amount of work has been accom- 
1 Hippocrates of Chios was one of the greatest geometers of antiquity ; 
he lost his property, as a merchant, by piracy or chicanery. Aristotle speaks 
of him as ‘‘slow and stupid.” _*‘ There seems to be no other ground for the 
criticism than that a Greek would call a mana fool who was cheated of his 
property. There are still extant mathematicians who are singularly deficient 
in ability for any studies but their own.” f 
2 In his “‘English Mathematical and Astronomical Writers” (companio1, 
to “‘ British Almanac for 1837,” p. 38), De Morgan made one of his shrewd 
guesses that Billingsley’s (first English) Euclid was certainly made from th= 
Greek, and not from any of the Arabico-Latin versions. This surmise has 
been found correct by G. B. Halsted in the American Journal of Mathe- 
matics, vol. ii. pp. 46-48. We notice that Mr. Gow gives the same reference 
in his Addenda. Mr. Gow gives a proof of a prop. (xxii.) of Euclid’. 
optics which recalls a passage in the recent brochure “‘ Flatland :” it is, 
“Ifa circle be described in the same plane as the eye, it will seem to be a 
straight line.” 
