contrived a machine which bears some resemblance to a modern steam-engine. In the fifty-sixth Article of his book he gives a brief account of a wheel which he invented and exhibited in the Tower before King Charles I. and several members of his court. 66 He says it was a most incredible thing if not seen"; and that "The wheel was 14 foot over, and 40 weights of 50 pounds apiece." It would seem to have been of the nature of the contrivance noticed in Art. 765. The other discovery is claimed by Orffyreus, who was born in Alsace. He is said to have constructed for the Landgrave of Hesse Cassel a wheel which revolved 25 or 26 times in a minute; and continued in motion for two months. A letter on this wheel is in print which was addressed by 's Gravesande to Newton, speaking favourably of the contrivance and its inventor. The most probable conjecture is that the rotation was produced by clockwork concealed in the wheel. Orffyreus is said to have destroyed his wheel in dissatisfaction with the treatment which he received.

770. Amusing instances are on record of the confidence of sanguine speculators in the success of their projects. Thus we find one person whose anxiety was as to whether he should ever be able to stop his machine when once in motion. Another person proposed a modification of the waterwheel and pump scheme of Art. 766, and being in alarm lest his machine should pump up more water than required for itself, suggested the use of a wastepipe by which the superfluous water could in fact be thrown away. In a case which was submitted to the late Dr Whewell the only misgiving of the projector arose from moral considerations; he feared that the labouring classes having no longer any necessity for toiling would sink into idleness and vice.

771. The history of the subject reproduces perpetually the same features. Accident brings the idea of perpetual motion to the attention of a man who can handle tools, and who has some inventive faculty, a quality by no means rare but which is of little use except it be combined with a knowledge of what has been done before. Such a person however is in general ignorant of all the preceding failures

and also of the principles on which a well-trained mathematician relies for his conviction of the impossibility of attaining the proposed result. He soon convinces himself that he has succeeded, and having probably exhausted his own funds he applies to some person who can furnish capital for constructing models and engines, and for making the invention known. Loss and perhaps ruin ultimately fall on the sanguine but undisciplined contriver and the credulous patron.

EXAMPLES.

IV. MOTION. FALLING BODIES.

1. A railway train performs a journey of 45 miles in 2 hours: find the velocity in feet per second.

2. A train is moving at the rate of 270 yards per minute: express this velocity in feet per second.

3. The distance of the Moon being about 240000 miles find the uniform velocity of a body which would pass from the Earth to the Moon in 400 days.

4. The minute-hand of a watch is twice as long as the second-hand: shew that the end of the second-hand moves thirty times as fast as the end of the minute-hand.

5. Find the space described in the fifth second by a falling body.

6. A body falls for six seconds: find the space in the last two seconds of the fall.

described

7. Find the space described by a falling body in one tenth of a second beginning at the end of four seconds.

8. Find the space described by a falling body in one twentieth of a second beginning at the end of two seconds.

9. If a body fall for a quarter of a minute shew that it would then be moving at the rate of 480 feet per second; and ascertain what this velocity will be, expressed in miles per hour.

10. Shew that a falling body acquires in the seventh of a second a velocity of about three miles per hour.

11. Find the velocity of a falling body at the end of two seconds, and also at the end of two seconds and a twentieth and shew that the space actually described in the twentieth of a second is the same as if the body had moved uniformly with a velocity equal to half the sum of the two velocities.

12. A stone dropped into a well is heard to strike the water in two seconds and a quarter: find the depth of the well.

13. Shew by Art. 88 that the spaces described by a falling body in the first, second, third, fourth... seconds are in the proportion of the successive odd numbers 1, 3, 5, 7,... 14. A falling body is observed to describe 336 feet in one second: find how long it has been falling altogether.

15. A body has been falling for a certain number of seconds: shew that the number of feet described in the next two seconds is the product of 64 into the number of seconds increased by unity.

16. A falling body is observed to describe 144 feet in two seconds: find how long it had been falling when it was first observed.

17. Shew by numerical examples that the velocity of a falling body at the middle of any interval is half the sum of the velocities at the beginning and the end of the interval. For instance, half the sum of the velocities at the end of three seconds and at the end of three seconds and a half is the velocity at the end of three seconds and a quarter.

18. Shew that the space described by a falling body between the end of three seconds and the end of three seconds and a half is the same as would be described by a body moving uniformly with the velocity which the falling body has at the end of three seconds and a quarter. Take other examples of a similar kind.

19. Two balls are dropped at the same instant from two different points, one vertically above the other: shew that the balls as they fall always keep at the same distance from each other.

20. Two balls are dropped from the same point at different instants: shew that as they fall the distance between them increases continually with the time.

V. RELATIVE MOTION. COMPOUND MOTION.

1. A steamer is moving at the rate of 20 feet per second: find the resultant velocity of a ball which is shot from the stern to the bow of the steamer with a velocity of 15 feet per second.

2. Find also the resultant velocity of the ball if it is shot from the bow to the stern.

3. Find also the resultant velocity of the ball if it is shot from side to side.

4. One steamer moves from West to East at the rate of 12 miles per hour, and another moves from South to North at the rate of 16 miles per hour: shew that one will separate from the other at the rate of 20 miles per hour: the steamers start from the same point at the same time.

5. An express train 66 yards long moving at the rate of 40 miles an hour meets a slow train 110 yards long moving at the rate of 20 miles an hour. Find how long a man in the express train takes to pass the slow train, and how long the express train takes in completely passing the slow train.

6. A railway train is moving at the rate of 60 miles per hour, and a ball is dropped from a point at the end of the train which is 16 feet above the ground: shew that if the ball did not partake of the motion of the train when it reached the ground it would be 88 feet behind the train.

7. A river one mile broad is running downwards at the rate of four miles an hour; a steamer can go up the river at the rate of six miles per hour; find at what rate it can go down the river.

8. A stone after falling for one second strikes a plane of glass in breaking through which it loses half its velocity: find how far it will fall in the next second.

9. Through what space must a heavy body fall from rest in order to acquire a velocity of 160 feet per second? If it continue falling for another second after having acquired this velocity find through what space it will fall.

10. A heavy particle is dropped from a given point, and after it has fallen for one second another particle is dropped from the same point. Find the distance between the particles when the first has moved for five seconds.