48 
Much of our polite learning has been in the possession of. the world 
for two thousand years or more, but the peculiarity of our present civili- 
zation is the general diffusion of knowledge and the triumphs of engineer- 
ing skill. Invention and machinery have multiplied man-power by twenty. 
And below it all lies the calculus. The successful engineer who would 
be anything but a mere slavish copyist must have a mind well founded in- 
mathematics. 
Let us refer briefly to some of the things which calculus has done or 
helped to do in engineering. We may say with little danger of contradic- 
tion that the engineering of to-day is a question of minimum causes and 
maximum effects—a question of the first differential coefficient. 
A Tay bridge disaster reveals a crime against humanity. We won- 
der at the pyramids and temples of Egypt, but when we think of the 
lives sacrificed like flies in these colossal but useless works, where the 
means employed were vastly out of proportion to the ends sought, here, 
too, was a crime against humanity. 
It is equally beneath the dignity of the disciplined man to put too 
much or too little into a structure to serve its designed purpose in use and 
beauty. 
In hydraulics, calculus investigates water pressure on a submerged 
surface and center of pressure for same, thus determining the size and 
form of retaining walls of all kinds, and solving the first problem of a 
water supply. It also investigates the quantity of discharge through ori- 
fices, notches and overweirs, determines the most economical sections of 
conduits and canals, the time of emptying or filling locks or other vessels 
under a varying head; the maximum range of jets from a given inclination 
determines empirical formulae from experimental data by the aid of least 
squares; discusses non-uniform flow in rivers and back water curve above 
dams; discusses the Maximum work derived from moving vanes, such as 
stationary water wheels, wheels of steam boats, and screws of propel- 
lers. 
Fifty years ago an excellent engineer by the name of Uriah Boyden 
spent weeks in designing the buckets of a water wheel. He obtained 
correct forms, but by the aid of the calculus a man no more talented 
naturally may to-day do the same work in two or three hours. 
In machinery and structures it investigates the work absorbed by 
friction of pivots and the like; moments of inertia and centers of gravity 
