44. Mr. M. The reference to the clock pendulum reminds me that our hour has elapsed, and I shall expect you to come next week with what you can prepare on the subject of MECHANICAL POWERS.

1 IM'-PULSE, force communicated.

2 DI'-A-GRAM, a figure drawn for the purpose of explaining some principle.

3 DI-VERG'-ENCE, a receding or separating from each other.

4 A'-RE-A, extent of surface.

5 DE-SCRIB'ED, passed over.

6 "The square" of a number means the product of that number by itself.

7 PROB'-LEM, some question in mathematics requiring a solution.

8 QUAD'-RU-PLE, increased four fold. 9 EX-HAUST', draw out; remove. 10 DE-TACH', let loose; set free. 11 PAR'-A-CHUTE (par'-a-shute).

12 A'-ER-O-NAUT, one who sails or floats in the air; a balloonist.

13 HOLD, the whole interior cavity of a ship below the lower deck.

14 GY'-RO-SCOPE, an instrument for illustrating the phenomena of rotation and the composition of rotations.

15 RE-SULT-ANT, that which results from the combination of two or more.

16 Ax'-18, that which passes through the centre of the wheel, and on which it revolves.

17 STA-BIL'-I-TY, strength to stand without being overthrown.

18 BAL'-LAST-ED, kept steady by ballast.

LESSON VII.

MECHANICAL POWERS.*

1. Mr. M. The topic for this lesson will certainly be an interesting one to the young ladies who wish to know about scissors and sewing machines, as well as to the lad who knows all about mills, and the one who understands the mechanical1 arrangements used in farm work; but as for Frank, who has spent his life thus far in his father's office and the Latin school, I can hardly expect that our mechanical lesson will be so pleasing to him.

2. Frank. But, Mr. Maynard, while I was reading Cæsar and Virgil, I found it necessary to know something about the mechanical powers, in order to understand the machines which the Romans used to batter down walls, and to discharge arrows, darts, and stones. I have constructed a model of Cæsar's bridge,2 from his description of it; and also models of the catapulta, ballista, and scorpio;3 and I think no one can feel more desirous to understand the mechanical powers than I do.

3. Mr. M. Very well; I am glad to see that you appreciate the importance of such knowledge to a correct under

* The Mechanical Powers are certain instruments or simple machines employed to facilitate the moving of weights, or the overcoming of resistance.

standing of what you read. What is a simple machine, Frank?

Frank. An instrument by which weights can be raised, resistance of heavy bodies overcome, and motion communicated to masses of matter.

4. Mr. M. A very good definition. How many primary mechanical powers are there?

John. Three; the lever, pulley, and inclined plane.

Mr. M. That is the division I prefer, as the wheel and axle, the wedge, and screw, are modifications of the first three. What is a lever?

George. A lever is an inflexible bar, supported on a point called a fulcrum," about which it moves freely.

5. Mr. M. I like to have you give the definitions so clearly. In the cut which I here show you, you see a man trying

Fig. 14.

to move a heavy stone. Here L is the lever, F the fulcrum, W the weight. By pressing down at the end L, the other end of the lever raises W, the weight. The centre of motion is at F, the fulcrum. In other words, the power or force resting on the prop or fulcrum overcomes the weight or resistance. Thus, if the lever be under the centre of gravity of the weight, and the length of the lever from the fulcrum be twice as long as the other part, a man can raise the weight one inch for every two inches he presses down the end of the lever.

6. I wish you to notice that there are four things to be considered, viz., the power applied, and its distance from the fulcrum; also the weight or resistance, and its distance from the fulcrum. Now if the stone weighs 500 pounds, and is two feet from the fulcrum, how much power must the man apply, at a distance of five feet from the fulcrum, in order to move the stone?

7. John. I have learned that in all such cases the product of the weight by its distance is equal to the product of the power by its distance; therefore I find the required power to be 200 pounds.

Ella. Please explain your work for our benefit, and not come to the conclusion so suddenly.

8. John. The weight 500, multiplied by its distance 2, is 1000. The product of the power by its distance must be equal to 1000. But the distance of the power is 5, hence the other factor, or the power, will be found by dividing 1000 by 5, which will give 200.

Ella. Are the calculations for all kinds of levers made so easily?

9. Mr. M. I am most happy to assure you that not only are all calculations pertaining to the lever thus simple, but also all calculations of the other simple mechanical powers. Do you understand this expression, P× Pd=WxWd?*

10. George. I think it must mean that the product of pow er by power's distance from the fulcrum is equal to the product of the weight by the weight's distance from the fulcrum.

11. Mr. M. That is the law for equilibrium; but to produce motion the power must exceed that necessary for equilibrium or balancing. Universally, the product of the power by the distance it moves is always equal to the product of the weight by the distance it moves in a vertical direction. Whenever you have any difficulty in solving questions in mechanical powers, think of this principle.

12. John. Does not the weight of the long end of the lever interfere with this rule? I saw some engineers once weighing the lever of a safety-valve, and heard them say the rule for calculating levers would not do for them.

Mr. M. Very true; the weight of the lever is a part of the power, and should be so calculated. In the formulaR I have given you the lever is considered as without weight.

13. John. As all levers do really have weight, will you please show us how to estimate that weight in practice? Mr. M. Have you not been able to find the information you seek in any school-books or mechanics' manuals ??

John. No, sir. I have searched diligently even in college text-books in your library.

14. Mr. M. I really can not point you to the book where

* This should be read, "P multiplied by Pd equals W multiplied by Wd."

you will find what you wish, and what is so important, but I think it can be made very plain. We will use this diagram for our illustration.

15. Suppose the lever to be a bar of iron sixteen inches long, every inch of which weighs one pound, and that the fulcrum, F, is six inches from the weight, W. The centre of gravity of the short arm will be three inches from the fulcrum, where the weight will be six pounds. The centre of the long arm will be five inches from the fulcrum, where its weight will be ten pounds. Now we have only to calculate the short end as an additional weight of six pounds three inches from the fulcrum, and the weight of the long arm as a power of ten pounds five inches distant, and combine these with the theoretical10 calculation.

16. John. I think I can now accomplish what I have heard many mechanics wish themselves able to do. The problem does not seem to be a very difficult one.

Mr. M. Will you tell me, then, with such a lever, what power at P will balance 100 pounds at W?

17. John. If we multiply 100 by 6 (six inches), we have 600. Then 6 pounds, the weight of the short arm of the lever, multiplied by 3 (three inches), will give 18, which, added to 600, will make 618, for the products of the weights by their distances. Then, for the long arm of the lever, we multiply the weight 10 by its distance 5, and take the product, 50, from 618, and this will leave 568 pounds to be balanced by a weight at P; but, as P is ten inches from the fulcrum, we divide 568 by 10, and this gives us 56 pounds and eight tenths of a pound.

18. Mr. M. You are correct in your answer. Fifty-six pounds and eight tenths of a pound at P will balance one hundred pounds at W. Can you tell me what would have been the theoretical answer?

Ida. I have already made the calculation, and I find, if we

suppose the bar or lever not to have any weight, 60 pounds at P will balance 100 at W.

19. Mr. M. Thus, you see, there is a difference of over three pounds. If the lever is not a straight and uniform bar, the distance of the centres of gravity of its arms must be calculated by means we can not introduce here.

Ida. I used to learn about three kinds of levers. Can the

power of all of them be calculated in the same way?

20. Mr. M. Yes. Their parts are essentially the same; viz., the power and its distance, and the weight and its distance from the centre of motion; and the formula I gave will solve them all. Can you tell me what constitutes a lever of the second kind?

21. George. The second kind of lever is that in which the weight and the power are on the same side of the fulcrum, and the power is furthest from the fulcrum.

Thus, if a mason desires to move forward a large piece of stone, instead of bearing down upon the lever to raise it up a little, he sticks his crowbar into the ground, and pushing upward, moves the stone little by little onward, the ground being the fulcrum.

Fig. 10.

22. John. Is not a common wheelbarrow a kind of lever of this kind?

Mr. M. It is a lever on a rolling fulcrum. So, also, is the oar of a boat, the water being the fulcrum, the person who rows the power, and the boat itself the resistance.

23. Frank. It seems to me that the masts of a ship are levers.

Mr. M. So they are; and also the rudders by which ships are steered. Can the young ladies give me some examples

of levers either of the first or second kind?

Ida. Nut-crackers and lemon-squeezers are levers of the second kind.

Ella. Scissors, forceps, and snuffers are double levers of the first kind.

24. Mr. M. Well said; for when you readily state the kind of levers, I think you understand what is the fulcrum, power, and weight. The scale-beam used in weighing is a simple