494 Proceedings of the Royal Society of Edinburgh. [Sess. 
have been answered out of Todhunter’s Elementary Algebra. . . . The 
problem for the writer of a text-book has come now, in fact, to be this — 
to write a booklet so neatly trimmed and compacted that no coach, on 
looking through it, can mark a single passage which the candidate for a 
minimum pass can safely omit. Some of these text-books I have seen, 
where the scientific matter has been, like the lady’s waist in the nursery 
song, compressed so 4 gent and sma’,’ that the thickness of it barely, if at 
all, surpasses what is devoted to the publisher’s advertisements. 
“ The cure for all this evil is simply to give effect to a higher ideal 
of education in general, and of scientific education in particular. . . . 
Science cannot live among the people . . . unless we have living contact 
with the working minds of living men. It takes the hand of God to make 
a great mind, but contact with a great mind will make a little mind 
greater. ... In the future we must look to men and to ideas, and trust 
less to systems. Systems of examination have been tested and found 
wanting in nearly every civilised country on the face of the earth. . . . 
The University of London . . . has for many years pursued its career as a 
mere examining body. It has done so with rare advantages in the way 
of Government aid, efficient organisation, and an unsurpassed staff of 
examiners. Yet it has been a failure as an instrument for promoting the 
higher education — foredoomed to be so, because, as I have said, you must 
sow before you can reap.” 
Within a year of uttering these stirring words, Chrystal presented to 
the mathematical world of teachers his solution of part of the difficulty in 
the form of the first volume of his well-known Algebra. Its merits 
were immediately recognised. The first object he set before him was “to 
develop algebra as a science, and thereby to increase its usefulness as an 
educational discipline.” The introduction into an elementary text-book of 
such subjects as the complex variable, the equivalence of systems of 
equations, the use of graphical methods, and the discussion of problems of 
maxima and minima, was a new feature; but subsequent developments 
have fully vindicated the innovations. Graphical methods of representing 
simple functions, and the transition to the solution of equations, are now the 
stock-in-trade of every modern Algebra. From these methods, the principles 
of co-ordinate geometry flow naturally and simply ; and Chrystal himself, 
somewhat later, made a further valuable contribution to the nomenclature 
of the subject by his introduction of the expressive terms, “ the constraint 
equation ” and “ the freedom equations of a curve.” The second volume 
followed in 1889, and was marked by the same lucid and logical treatment 
of the more advanced parts. In 1898 Chrystal further enriched the 
