P. HISTORY OF ' AATHEMATICAL - School of Mathematics

256 A HISTORY OF MATHEMATICAL NOTATIONS
Chuquet h ds two mean proportionals between 8 and 27 by the
scheme
" 8 27
where 12 and 18 are the two mean proportionals sought; i.e., 8, 12, 18,
27 are in continued proportion.
221. Proportions involving fractions.-Lines forming a cross ( X ),
together with two horizontal parallel lines, were extensively applied
to the solution of proportions involving fractions, and constituted a
most clever device for obtaining the required answer mechanically.
If it is the purpose of mathematics to resolve complicated problems
by a minimum mental effort, then this device takes high rank.
The very earliest arithmetic ever printed, namely, the anonymous
booklet gotten out in 1478 at Treviso,' in Northern Italy, contains an
interesting problem of two couriers starting from Rome and Venice,
respectively, the Roman reaching Venice in 7 days, the Venetian
arriving at Rome in 9 days. If Rome and Venice are 250 miles apart,
in how many days did they meet, and how far did each travel before
they met? They met in 3)g days. The computation of the distance
traveled by the courier from Rome calls for the solution of the proportion
which we write 7: 250 = f: : x.
The Treviso arithmetic gives the following arrangement:
The connecting lines indicate what numbers shall be multiplied together;
namely, 1, 250, and 63, also 7, 1, and 16. The product of the
latter-namely, 112-is written above on the left. The author then
finds 250>