wheel whose rim is formed of two metals of different expansive powers, so arranged that the change of size of the wheel, as the temperature rises or falls, is compensated for by the change in position of the parts of the rim.

The anchor escapement was employed in that popular and excellent timepiece used throughout the eighteenth and in the early part of the nineteenth century and now known as the Grandfather Clock. In this clock the pendulum is hung from a strip of thin steel spring, which allows it to oscillate and supports it without friction. This manner of supporting pendulums is now very much in use.

The watch differs from the original clock in that it has a vibrating wheel instead of a vibrating pendulum. As in a clock gravity is always pulling the pendulum down to the bottom of its arc, but does not fix it there, because the momentum acquired during its fall from one side carries it up to an equal height on the other, so in a watch a spring, generally spiral, surrounding the axis of the balance-wheel, is always pulling this toward a middle position of rest, but does not fix it there, because the momentum acquired during its approach to the middle position from either side carries it just as far past on the other side, and the spring has to begin its work again. The balance-wheel at each vibration allows one tooth of the adjoining wheel to pass, as the pendulum does in a clock, and the record of the beats is preserved by the wheel which follows. A main spring is used to keep up the motion of the watch, instead of the weight used in a clock, and as a spring acts equally well whatever be its position, the watch keeps time altho carried in the pocket or in a moving ship. In winding up a watch one turn of the axle on which the watch is fixed is rendered equivalent by the train of wheels to about 400 turns or beats of the balance-wheel, and thus the exertion during a few seconds of the hand which winds up gives motion for twenty-four or thirty hours.

The laws of the mechanism of the clock can easily be understood. The experiments with the pendulum and

with springs revealed certain principles which were early reduced to six and can be stated thus:

(1) A harmonic motion is one in which the accelerating force increases with the distance of the body from some fixed point.

(2) Bodies moving harmonically make their swings about this point in equal times.

(3) A spring of any sort or shape always has a restitutional force proportional to the displacement,

(4) And therefore masses attached to springs vibrate in equal times, however large the vibration may be.

(5) The bob of a pendulum, oscillating backward and forward, acts like a weight under the influence of a spring and is therefore isochronous.

(6) The time of vibration of a pendulum is uninfluenced by changes in the weight of the bob, but is influenced by changes in the length of the pendulum rod. The time of vibration of a mass attached to a spring is influenced by changes in the mass.

Early attempts were made to use a pendulum clock at sea, suspending it so as to avoid disturbance to its motion by the rocking of the ship. These proved vain. It therefore became desirable that a watch with a balance-wheel should be contrived to go with a degree of accuracy in some respects comparable with the accuracy of a pendulum clock. To encourage inventors an Act of Parliament was passed in the thirteenth year of Queen Anne's reign promising a large reward to any one who would invent a method of finding the longitude at sea true to half a degree that is, true to thirty geographical miles. If the finding of the longitude were to be accomplished by the invention of an accurate watch, then this involved the use of a watch that should not, in several months' going, have an error of more than two minutes, or the time the earth takes to turn through half a degree of longitude.

This was the problem which John Harrison, a carpenter of Yorkshire, made it his life business to solve. His

efforts lasted over forty years, but at the end he succeeded in winning the prize. His instruments have been much improved by subsequent inventors and have resulted in the construction of the modern ship's chronometer, a large watch about six inches in diameter, mounted on axles, in a mahogany box. The marine chronometer differs from the ordinary watch in the principle of its escapement, which is so constructed that the balance is free from the wheels during the greater part of its vibration, and also in being fitted with a compensation adjustment similar to that in the balance-wheels of the finer clocks and watches. The balance-spring of the chronometer is helicoidal, that of the watch spiral.

One of the inventions of modern times is the pneumatic clock, which is one of a series of clocks governed by pulsations of air sent at regular intervals to them through tubes by a central clock or regulator. The movement of the central clock compresses the air in the tube and causes a bellows to expand on each dial, thus moving the hands. Another recent invention is a clock without wheels or pendulum. It consists solely of two inclined plates with zigzag tracks and the clock framework supporting them. A perforated disk connected with the shaft which journals in the frame and two ball weights suspended in each tower and connected by means of a cord to the shaft successfully furnish the motive power. These weights are raised daily.

So the ingenuity of man goes on measuring this earthly element of time. Laplace said that "Time is to us the impression left on the memory by a series of events," and that motion, and motion only, can be used in measuring it. Thus it is motion, whether of the shadow on the grass, the dropping of water or the continuous oscillations of a swinging body, which is the necessary and unvarying element in all the measurements of time.

CHAPTER III

SURVEYING AND NAVIGATION

ONE of the earliest necessities of civilization was a system of ascertaining by measurement the shape and size of any portion of the earth's surface and representing the results on a reduced scale on maps. This is the surveyor's art and is supposed to have originated in Egypt, where property boundaries were annually obliterated by the inundations of the Nile. In Rome surveying was considered one of the liberal arts, and the measurement of lands was entrusted to public officers, who enjoyed certain privileges.

Julius Cæsar conceived the idea of a complete survey of the whole empire. For this purpose three geometers were employed: Theodotus, entrusted with the survey of the northern provinces; Zenodoxus, with the survey of the eastern, and Polycletus, of the southern. It is stated that a partial survey was finished 19 B.C. and the whole completed in 6 A.D. The materials collected were lodged in the public archives, receiving from time to time marks and notes to designate the various changes in the provinces. It was consulted by Pliny. The numerous changes at length required the construction of another chart with corrected measurements, which was effected about 230 A.D. under Alexander Severus. Of this chart the celebrated document Tabula Peutingerianæ is supposed by some modern critics to be an imperfect copy.

The mathematicians of the Alexandrian school made a distinct contribution to the art of surveying. Most authori

ties believe Heron of Alexandria to be the author of "Dioptra," tho some writers have attributed it to another mathematician of a later date by the name of Heron. "Dioptra," says Venturi, "were instruments resembling the modern theodolites. The instrument consisted of a rod, four yards long, with little plates at the end for aiming. This rested upon a circular disk. The rod could be moved horizontally and also vertically. By turning the rod around until stopped by two suitably located pins on the circular disk, the surveyor could work off a line perpendicular to a given direction. The level and plumb line were also used." Heron explains, with the aid of these instruments and of geometry, a large number of surveying problems, such as to find the distance between two points, only one of which is accessible, or between two points which are visible but both inaccessible; from a given point to run a perpendicular to a line which cannot be approached; to find the difference of level between two points, and to measure the area of a field without entering it. The "Dioptra" discloses considerable mathematical ability, but it gives rules and directions without proof.

The higher development of the art of surveying, like so many other mechanical arts depending on mathematics, is of comparatively recent date. The enormous areas of new land opened for habitation in the New World, the construction of railroads, bridges and water works have employed the keenest practical minds in solving large surveying and engineering problems, of which the Government does a large part.

Surveys may be divided into three classes: First, those made for general purposes, or information surveys, which may be exploratory, geodetic, geographic, topographic or geologic; second, those made for jurisdictional purposes, or cadastral surveys, which define political boundaries and those of private property and determine the enclosed areas; third, there are surveys made for construction purposes, or engineering surveys, on which are based estimates of the