may be secured with as little loss, both of time and money, as the nature of the business will admit; and likewise to allow time for the money to be paid, leaving it to the choice of the patentee whether he will pay the money immediately at the conclusion of the business, or at the expiration of (say) six months, the latter method being subject to an addition equivalent to the time allowed, and in case of non-payment, the invention to become the property of government.

In this age of invention, some arrangement of the kind is absolutely necessary, to put the poor man of genius somewhat on an equality with his more wealthy competitors.

In hopes of seeing the subject discussed in your interesting pages, I am, Sir,

Worfield.

Your's, &c. PERSEVERENTIA.

PILE DRIVING.

Bulbourne, April 3, 1827. Sir, In the 183rd and 185th Numbers of the Mechanics' Magazine, I find that two of your correspondents, "Glevum," and "William Andrews," charge me with having given an erroneous answer to the question on Pile Driving. I, therefore, in justice to myself and the subject, claim a small space in the pages of your valuable publication, in order to vindicate myself from this very serious charge.

As plain as I could possibly express myself, I said that the rule given was a rule for finding the comparative forces with which a ram strikes a pile, after falling from rest, through any given spaces. I never designed it as a rule for finding the absolute forces. After having calculated the forces in the assumed examples, according to the rule, I said that their forces were equal to those exerted by bodies of the weight determined by the calculations, after falling from rest through the space of one foot. But, perhaps, your correspondents may ask, what I mean by a rule for finding the comparative forces? By a rule, then, for finding the comparative forces, I mean such an one as will shew the proportion which the forces exerted by rams of different, or of the same, weight, after falling through any giving spaces, bear to each other. One or two examples will

make this quite plain. Suppose it be asked, which of two rams, one of which is 2 cwt., and has to fall from rest through the space of 36 feet, and the other of which is 6 cwt., and has to fall from rest through the space of 6 feet, strikes the pile with the greater force? According to my rule, we have 2 × √36-12, and 6x/9=18, for the comparative forces. Whence we learn that the latter exceeds the former in the proportion of 18 to 12, or of 3 to 2, that is, the force exerted by the greater ram, is half as much more as that exerted by the smaller one. Again, suppose it be asked, how many times greater the force exerted by a ram of 6 cwt., after falling from rest through the space of 20 feet, would be than the force exerted by the same ram after falling from rest through the space of one foot? According to the above named rule, we have 6×20=26.82, and 6 x √/1=6, for the comparative forces. Whence we learn that the force exerted by the ram after falling through the space of 20 feet, is 447 times greater than the force exerted by the same ram after falling through the space of one foot. Now these questions are correctly answered by my rule, which depends only upon the well known laws of percussion and falling bodies, that the force of percussion is as the weight of the porcutient body multiplied by the velocity, and that the velocity of falling bodies is as the square root of the space fallen, and proves it to be strictly correct for what it was intended. In order to get the absolute forces, we have only to multiply the comparative ones determined by my rule, by some practical factor which must first be determined by experiment. Thus, if it be true, (which I am not certain is the case) that the effective force, or momentum, of a body falling freely (from rest) through the space of 14 inch, is doubled," this factor will be 6.2. It is shown in my former letter on this subject, that the comparative force of the ram of 6 cwt.after falling 20 feet, is 6 X √20= 26.82; this, therefore, multiplied by the practical factor 6.2, gives 166.28 cwt., for the absolute force.

What "Glevum" means by saying that my calculation" is not in accordance with my own rule," I own, is past my comprehension, and I am quite sure it is beyond his power to prove how it can be otherwise to be in accordance with it. The rule directs to multiply the weight of the ram by the square root of the space fallen, and this is done; where, then, is the inconsistency? But, enough on this part of the subject. I shall now

proceed to show, that, however maturely he may have considered the question, and however particular he may have been "to guard against the propagation of error," "Glevum's" rule for calculating the force when the engine is declined, is absolutely erroneous. He says, when the engine is declined, "the perpendicular height only must be calculated upon," which, (supposing the perpendicular height at which the ram is disengaged to remain the same) is virtually asserting, what "Aries" has done, that "the declination of the engine will neither increase nor diminish the force." It is certainly true, that "the perpendicular height only must be calculated upon;" but this is not all; part of the absolute weight of the ram must be deducted, as shown in my former letter on this subject. It is there shown that the effective part of the weight of the ram, acting in the direction of the engine, is, (in the proposed question) but 525 cwt.; this, therefore, multiplied by the square root of 20, and then by the practical factor 6.2, gives 145.5 cwt. for the absolute force, whereas, according to "Glevum's" mode of calculating, it would be 166.26 cwt., which is too great by 20.76 cwt.!! There are also other cases in which the results obtained by "Glevum's" formula are equally erroneous. Thus, suppose it be asked with what force a ram of 6 cwt. would strike a pile after falling the 100th part of an inch; according to the formula M=va we have 18 for "the momentum," which, multiplied by 6, the weight of the ram, gives 1.08 cwt. for the absolute force, which is nearly 5 cwt.. less than the force exerted by the body in its quiescent state, that is, before it began to move at all!!!

4 s

4x.01 1.25

"Wm. Andrews," in his calculations of the force of a ram of 2 cwt. after falling through the space of 9 feet,' misrepresents my rule. For every one who has read my former letter must see that the conclusion which he draws is contrary to that rule. He calculates the comparative force as directed in my letter, and then asserts that I should call this force equivalent to the force exerted by a body of 6 cwt. in its quiescent state, whereas every one who has read that letter knows that I should say that this force was equal to the force exerted by a body of 6 cwt. after falling from rest through the space of one foot. Whether this "would not be sufficient to drive a pile a foot square through gravel or strong clay," I leave your readers to judge.

After reading the part of W. A.'s

271

letter above alluded to, I must confess, I was not a little surprised to meet with the following quotation from Dr. Walker. He writes, "I find Dr. Walker says, if a battering-ram be 1000lb. weight, and the velocity with which it strikes a wall be 20, then is its momentum 20,000; but a cannon-ball shall do the same execution if its quantity of matter be no more than 10lb., provided it be thrown with a velocity equal to 2,000, for 10 × 2,000 20,000; that is, the momentum is equal to the body, multiplied nto the velocity." Now, the rule given in the latter part of this quotation is exactly the same as the one given in my former letter; for I there said, that the force of percussion is measured by the product arising from the body moved multiplied by the velocity, which is exactly the same as the one quoted by W. A. from Dr. Walker, in order to prove my rule to be erroneous! But I would ask "W. Andrews" what is meant by the very indefinite expression, "a velocity of 20?" The fact is, the rule given by Dr. Walker is only a comparative one, and we learn nothing by it till we compare the forms of different weights moving with different velocity. Thus, in the example of the battering-ram and cannon-ball, we learn nothing but this-that a body of 1000lb. weight moving with any velocity would strike an object with no greater force than a cannon-ball of 10lb. weight moving with a hundred times greater velocity.

I trust I have now proved to the satisfaction of every one of your readers, that "Glevum" and W. Andrews have asserted what they have not proved, namely, that the rule given by me is erroneous. I am, Sir,

Your obedient Servant,

WM. LAKE.

P.S. There are some typographical errors in my former letter, which I did not think worth the trouble of a letter to correct. The most material is in the rule, where part of a sentence is omitted. For, "by the square root from which it falls," read, "by the square root of the height from which it falls. The letters, also, in the Figure are wrong placed, but these may be easily corrected by any one who will take the trouble to read the part relating to it.

ON THE SECTOR, BY MONAD. The Line of Chords. (Continued from page 190.) The method of taking the measure of angles already given is an evident

converse of the last. angular point A (see Fig. col. 1, p. 199) with any radius A P describe an arch cutting both legs of the angle. Make A Pa parallel between 60. and 60 of the line of chords. Extend the compasses to the distance included between the points P Q where the arch cuts the legs of the angle. Apply them to the line of chords so that this extent is a parallel distance

cases.

between some two corresponding points of the scale. Then the numbers annexed indicate the measure of the angle in degrees. On the plain scale the line of chords is continued up to 90 deg. ; and the method of protraction is somewhat different. The radius, or chord 60, being here invariable, we are to make A P=ch. 60 of the scale, in both the above Then the distance P Q being measured from the O point of the scale, will shew the measure of the angle and vice versa. From which process it is plain, that the sectoral chords give the readiest mode of protraction, when both scales are equally accurate;-but since the sectoral line has commonly a radius of nearly 6 inches, while that of the plain scale cannot, for convenience of practice, be carried far beyond 3 inches altogether, it is equally plain that we may expect greater correctness in the measure of our angles, as well as greater dispatch in using the sector. And this remark applies with equal force to all those lines which are truly sectoral. The line of rhumbs is also a line of chords, adapted to a division of the circle into 32 equal parts, answering to the points of the compass, and serves to protract the bearings taken by that instrument. The method is the same as the foregoing. When the angle is greater than 60 deg. if we use the sector, greater than 90 deg. or 8 points, if we use the plain scale, we are first to lay off the arch 60 deg., 90 deg., or 8 points, as the case may be, and then apply the rule to find the difference. To give one example, let the angle be 75 deg. which is to be protracted. Strike an arch as before, and lay off on the circumference the radius chord 60 deg. of the scale, and add it to the former

arch. On the plain scales the method is similar. The converse, or to find the measure of an angle greater than the extent of the scale, is too plain to require a comment. To inscribe a polygon of a given number of sides in a circle, let it be a polygon of 17 sides. The circle being measured by 360, therefore the quotient of is the measure of the arch which This quotient in degrees is 21 deg. subtends one side of the polygon.

360

17

3,

11"

17

One of

10'. 35". 17 17. Since =8 X take 8 times this quantity, or 169 deg. 3"" 21 Subtract 60 deg. 24.42". twice, and the remainder is 49 deg. 24.42′′.25" This is very little less than 49 deg. 25'. Therefore, on the take the arch 120 deg.; add to it circumference of the given circle, the arch 49 deg. 25'; and divide the whole thus obtained by a double bisection into 4 equal parts. these parts carried continually round the circumference, will give the angular points of the polygon without sensible error in circles of small radius. The reader, however, must not expect that he will succeed in his first trial of this and similar constructions; as they suppose him to have already acquired the habit of estimating small distances with considerable correctness by the eye, and this must necessarily be the result of practice. The reason for multiplying the expression for the side, (in this instance by 8, which is the most convenient number,) is, that the unavoidable error in taking the chord may be reduced as low as possible, and be nearly equally divided among all the sides.

(To be continued.)

NOTICES TO CORRESPONDENTS.

Mr. Nottingham's explanatory letter week, but shall appear in our next. Acwas received too late for insertion this knowledgments to other correspondents

are deferred for want of room.

Communications (post paid) to be addressed to the Editor, at the Publishers', KNIGHT and LACEY, 55, Paternoster-Row, London. Printed by Milne, Duckworth, & Co. 76, Fleet-st.