said that by subsequent improvements in both lenses and scale adjustment we can today read these instruments to within one-half second, or the angle made by our pencil at a distance of 11,000 feet. However, our bridge maker will get along well enough if his instrument reads to 10 or even 20 seconds, so he need not concern himself about the delicate refinements which the astronomer has attained.

Another difficulty experienced by the astronomers was in determining just where the center of the field of observation was while they were measuring their angles. Picard, toward the close of the seventeenth century, solved this problem by adding those invaluable aids to the surveyor, the spider-web lines crossed so that the exact center of the field is always before the eye. Picard, by the way, was the first one to make a really accurate measurement of the distance embraced in a degree of the earth's surface.

One of the great troubles with telescopes up to about the middle of the eighteenth century was that the glass was not suitable for making lenses. In fact until the chemists a long time after this stepped in and corrected the trouble, the glass was imperfect unless exactly the proper kind of sand was used in making it. But even where good glass was available, if it was in the form of a lens, there was trouble because the light would not pass through it without separating into its component colored rays. One of the mathematicians of the time demonstrated mathematically that this defect could be corrected in spite of the famous Newton's statement to the contrary. Dolland in 1758 made a series of experiments in which he found that by combining flint and crown glass he could correct all the light troubles of this sort.

In another place, the compass, together with the discoveries and calculations which have made it an invaluable aid to the engineer are discussed.

It was not until after the telescope had taken on much of its present refinement that engineering began to have a place in the exact sciences. From the dawn of history there have been notable engineering feats in almost every country, but they were largely conducted much as the Roman tunnel already mentioned, in that they were performed almost regardless of cost, especially labor cost. What science has done for engineering is to give it a chance to do the work with the least effort and cost. To be able to determine dimensions exactly is one of the first requisites of economy in engineering.

This example of the development of the surveyor's transit is only one among the many that might be used to show the influence of science on the engineer's instruments and tools. The engineer draws from all sciences and arts. He must have a dependable tape when he is making very accurate measurements. The ordinary woven tape is likely to stretch and there is no natural material at hand which is just suited for such a devise. The metallurgist and the physicist together have found that by combining steel with nickel and manganese almost an ideal material for a tape is found. It expands and contracts due to heat and cold only about one-millionth of its length; it is tough, rust proof, and otherwise dependable under reasonable care.

The engineer wishes to place a bridge pier in the middle of a river and must sink a hole to bed-rock to get a firm foundation. He could put a water-tight casing down, pump the water out and let the men work at the bottom. But experimentalists have found that it is unnecessary to pump the water out. Air under pressure will hold the water back; hence large air compression engines are installed. From results worked out in laboratory experiments, the engineer can calculate just how much pressure will be exerted in holding out the water and whether the men can work under such pressure. The

same law of the pressure of gases is used in several engineering tools such as the pneumatic hammers, certain water meters, drills, pumps, etc.

In the devices for measuring the strength of the materials with which he builds, the engineer must deal in tremendous pressures. In order to measure such pressures, the laws of the multiplication of forces by means of levers, stated by the old Greek experimenter and philosopher Archimedes, must be known. These same laws are used in the engineer's jack-screws, cranes, and similar implements.

Besides the instruments the engineer uses himself, there are many others which he uses indirectly through the hands of specialists in the different sciences. The meteorologist furnishes rain, frost, and wind data; the microscopist studies the minute structure and looks for the causes of the deterioration of materials of construction; the physicist studies the expansion and contraction of the various metals used; and the astronomer keeps track of the magnetic pole for him.

Thus we see that the engineer's instruments are almost wholly adaptations from the different sciences. The engineer's success is dependent upon his ability to do with one dollar what any bungler could somehow accomplish with two dollars. It is because he makes use of the exact sciences that he is able to get better results than the average person. The better the engineer the more he can see in the practical application of the newer sciences to help him with his problems.

The engineer must not, however, take all the credit for the marvelous things he accomplishes. He should recognize the foundation laid by such pioneer scientists as Tycho Brahe, Kepler, and the other silent, and largely forgotten workers who each added his bit to help harness the forces of our world. These quiet investigators did not as a rule reap direct returns for their labors, and

we are prone to judge men largely by their practical accomplishments, but the fruit time of industry would not have been possible except for the work of those who planted the seeds of research and left other generations and other workers to reap the results of their plantings.

CHAPTER XXXI

LEARNING HOW TO CALCULATE

What a catastrophe it would be should the steam engine, the electric motor, the telegraph and the telephone be removed on a moment's notice. Yet this disaster would be mild compared with the elimination of the world's knowledge of mathematics. On every side we meet something that has needed to be solved by calculation. Some of these problems have consisted of simple addition, subtraction, multiplication and division, but to a greater extent than one would suspect, the complicated and recently developed forms of mathematics such as calculus are used in constructing the ordinary articles around us. From the tiny pasteboard box to the huge water tank, from the thread spool to the locomotive boiler; the chances are the mathematician has had a hand in designing the article in order to make the most economical use of materials and space. Not many years ago, practically everything was made by guessing at the proper proportions; but if, for instance, you were making tin cans to hold a certain quantity of tomatoes, it might take dozens of guesses to determine the proper relation of the sides and the bottom that would give the least waste of tin, whereas the mathematician can with a very little figuring tell the exact relationship. Mathematics is the science which shows an exact way to make a short cut.

Most of us are interested in mathematics chiefly as a means for figuring our simple accounts; to determine our debits and credits. This represents about the highest