STUDY OF EUCLID IN THE MIDDLE AGES 367 
less admirable it is not clear that he actually found these 
things, and it is still less certain that the geometrical matter 
referred to was Boetius’s Geometry. 
From Gerbert’s time, again, no further progress was made 
until translations from the Arabic began with Athelhard and 
the rest. Gherard of Cremona (died 1187), who translated 
the Elements and an-NairizI’s commentary thereon, is credited 
with a whole series of translations from the Arabic of Greek 
authors; they included the Data of Euclid, the Sphaerica of 
Theodosius,the Sphaerica of Menelaus, the Syntaxis of Ptolemy ; 
besides which he translated Arabian geometrical works such 
as the Liber trium fratrum, and also the algebra of Muhammad 
b. Musa. ! One of the first results of the interest thus aroused 
in Greek and Arabian mathematics was seen in the very 
remarkable works of Leonardo of Pisa (Fibonacci). Leonardo 
first published in 1202, and then brought out later (1228) an 
improved edition of, his Liber abaci in which he gave the 
whole of arithmetic and algebra as known to the Arabs, but 
in a free and independent style of his own; in like manner in 
his Practice.i geometriae of 1220 he collected (1) all that the 
Elements of Euclid and Archimedes’s books on the Measure 
ment of a Circle and On the Sphere and Cylinder had taught 
him about the measurement of plane figures bounded by 
straight lines, solid figures bounded by planes, the circle and 
the sphere respectively, (2) divisions of figures in different 
proportions, wherein he based himself on Euclid’s book 0 n the 
divisions of figures, but carried the subject further, (3) some 
trigonometry, which he got from Ptolemy and Arabic sources 
(he uses the terms sinus rectus and sinus versus); in the 
treatment of these varied subjects he showed the same mastery 
and, in places, distinct originality. We should have expected 
a great general advance in the next centuries after such a 
beginning, but, as Hankel says, when we look at the work of 
Luca Paciuolo nearly three centuries later, we find that the 
talent which Leonardo had left to the Latin world had lain 
hidden in a napkin and earned no interest. As regards the 
place of geometry in education during this period we have 
the evidence of Roger Bacon (1214-94), though he, it 
is true, seems to have taken an exaggerated view of the 
incompetence of the mathematicians and teachers of his