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INSTRUMENTAL ARITHMETIC.

DESCRIPTION AND

USE OF AN IM PROVED SLIDING-RULE, ADAPTED TO ALL PURPOSES OF COMMON ARITHMETIC. "BY THOMAS SADDINGTON.

Sir, Some years ago I devised or invented (if you will allow the term) several very important improvements in the common decimal or Gunter's sliding-rule, which I have denominated the Improved Sliding-Rule; and I have now drawn up a concise description and illustration of its principles and advantages over the decimal rule, in all departments of common arithmetic. The description is adapted to a rule six inches long, which is a convenient size for the pocket; yet a rule of two feet long will be more suitable for the desk, and solve the questions with fourfold greater accuracy. As it is an original invention, and has never been made public, you will probably consider it deserving a place in your Magazine. It is my intention (on some future occasion) to submit to your readers a few short and plain directions for laying down the lines of a sliding-rule of any given length, by means of a table of logarithms.

Description.

The sliding-rule, containing a line of numbers laid down in decimal divisions, by means of logarithms, and generally called Gunter's line, is an instrument well known amongst mechanics.

The advantages derived from the improved sliding-rule, over the one in common use, consists in being divided duodecimally as well as decimally, whereby it is rendered applicable for working and solving nearly all questions in arithmetic, in pounds, shillings, pence, and farthings, (in proportion to the size of the instrument) by one operation, which cannot be done simply by the decimal line of numbers.

As this communication will probably fall into the hands of many persons unacquainted with the nature

and use of the common sliding-rule, it ta will be proper to make a few generali observations, illustrative of its con2 m struction, when applied to plain orǝd integral numbers.

On one side of this rule, which iss six inches long, (and may be termed 11 a pocket-rule) are laid down a double radius of the line of numbers marked~} N. This line contains the logarithms of the number, or, more properly, the i number corresponding to the loga! rithm, and is figured thus near the li left hand end of the rule is 1; and towards the right hand is 2, 3, 4, 5, 6,† 7, 8, 9, and then 1 in the middle, going still on 2, 3, 4, 5, 6, 7, 8, 9, and 10 at the end.

The first 1 at the left hand may be counted for 1, or 10, or 100, or 1000, &c., and then the next 2 will be 2, or 20, or 200, or 2000, &c.; so that if the first 1 be esteemed 1, the middleb 1 will be 10, and 2 towards the right! hand will be 20-3 will be 30, and 10 at the end will be 100. Again, if the first 1 is counted 10, the next 2 will be 20-3 will be 30, and 1 in the middle will be 100; the next 2 wills be 200-3 will be 300, and 10 at the end will then be 1000. In like manner, if the first 1 be estimated one tenth part, the next 2 is two-tenth parts, and the middle 1 is then 1, and the next 2 is 2, and 10 at the end will be 10.

As the figures are increased or decreased in their values, so in like manner must all the intermediate strokes or subdivisions be increased or diminished; that is, if the first 1 at the left hand be counted 1, the 2 on the right hand of it is 2, and each long stroke or subdivision between them is one-tenth part, and the short strokes between the subdivisions make twentieth parts. Between 2 and 5 are ten strokes, each being one-tenth part. Between 5 and 1 in the middle are five strokes, each being onefifth, or two-tenth parts; the 1 in the middle will now be counted as 10, the next 2 will be counted 20, and each long stroke will be 11, 12, 13, 14, 15, 16, 17, 18, 19; and 2 for 20, and so on to 5, which will be 50; from 5 to 10 there are only five strokes between each figure, which are counted 52, 54, 56, 58, and 6 for 60, and so to 10

at the end, which will be counted 100. If 1, either at the left hand or in the middle, is counted 100, 2 will be 200, 3 will be 300, &c., to 1 or 10, which will be 1000, and between 1 and 2 each long stroke will be ten, and the short stroke five. Between 2 and 5 each stroke will be ten, and between 5 and 1 or 10 each stroke will be twenty. If 1 is counted for 1000, then each of the foregoing must be increased in a tenfold proportion; and if 1 is counted for 10,000, then each must be increased an additional tenfold, and 2 will be 20,000, and so on to 1, which will be 100,000, &c.

From this short description it will not be very difficult to find the division representing any particular number, and as the figures and lines on each radius are exactly alike, either half of the rule may be used. The double set being laid down for the purpose of solving questions in proportional arithmetic, which cannot be done in most cases with only one set of figures and divisions. The figures on the stock or fixed part of the rule, and on the slide, are also exactly alike.

Supposing the point representing number 12 were required, take the division at the figure 1 in the middle and call it 10; then count two-tenths or long strokes towards 2 on the right hand, and that division will be the point representing the number 12. Again, suppose it is required to find the number 22, call the figure 2 for 20, then two strokes towards 3 will be the required point. Again, required 55, call 5 for 50, and two strokes and a half towards 6, between the second and third strokes, will be the required point. If 1 in the middle had been called 100, then 12 would have been 120, and 22 would be 220, and 55 would be 550.

Before I proceed to illustrate the advantages of the common slidingrule, when applied to plain or integral numbers, and the improved slidingrule in solving all questions in common arithmetic, it will be necessary to make two general observations, deduced from mechanics and optics, and which must be kept constantly in view throughout the whole of the following communication, in order to prevent any disappointment in the

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expectations of the young practitioner.

First. What you gain in power you lose in velocity, and what you gain in velocity you lose in power, that is to say, as the principle of one increases, the other decreases. Or, perhaps, secondly, the idea intended to be conveyed may be better illustrated from optics than mechanics, by the use of the telescope or spy-glass, for the apparent magnitude of an object and distinctness of vision are in proportion to the distance from the observer. An object distinctly seen at a given point will not be seen with the same degree of magnitude at a tenfold distance, and it would be seen with still less accuracy if removed to one hundred times the first distance from the ' observer. The sliding-rule rests on a similar principle, as will be easily seen by examining the proportionate distance of figures laid down upon it. The higher the numbers go, the less space is allowed for the respective divisions the same space that is allowed from 1 to 2 is only allowed from 4 to 8, and the same from 8 to 16. Again, the space given from 1 to 10 is only allowed from 10 to 100, and also the same from 100 to 1000; so that it will be readily granted, that the higher the numbers are, the less accurate will be the estimated value of your calculations; but in descending from high numbers, the accuracy is increased in the ratio of the two extremes, in the same manner as the observer of an object with a spy-glass, on moving nearer to it, in a tenfold proportion, would view it with an increased magnitude in the ratio of the decreased distance

Note. In using the sliding-rule, it is not intended that it should supersede the use of figures in the common mode of making calculations where accuracy is required, but the utility of the rule will be fully acknowledged as a corrector in all operations, in proportion to the size of the instru ment, for, by adjusting the slide to the respective questions, the solution is instantly obtained, whereby the calculator will be enabled to judge if his computations are correct, if not, the error, if considerable, will be immediately detected. Where great

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accuracy is not necessary, and only the proximate value or amount of any question is required to be instantly ascertained, or in working of low numbers, much gratification will result from the use of this instrument.

I shall now proceed with the illus tration, in working a few examples in multiplication, division, and the rule of three, which is the ground-work of every other rule of proportion, in arithmetic.

Multiplication.

Multiplication is performed by placing the figure and division denoting the multiplier, on the slide under unit, or 1, on the stock or fixed part of the rule and counting on towards the right hand, or back towards the left hand, until you come to the figure or division denoting the multiplicand, and under which you will have the product or sum total on the slide.

Examples.

1st. Multiply 4 by 2.

Here 2 is the multiplier; place the figure 2 on the slide, under 1 on the stock; by drawing the slide out towards the right hand, (or putting it back towards the left hand) until 2 on the first radius or half part of the slide comes under 1 on the middle of the stock, (or if the first 2 is made use of to be under 1 at the left hand) from thence count onwards to the right hand, until you come to the figure 4 on the stock, under which, on the slide, you have the figure 8 for the product required.

2nd. Multiply 15 by 2.

Keep the slide in the same position as before, and call 1 on the middle of the stock for 10, and count five of the long strokes which will be 15, and under that division on the slide is the figure 3, which now denotes 30 for the answer, the 1 having been increased tenfold, and called 10.

3rd. Multiply 45 by 2.

Keep the slide in the same position as before, and count on to 45; viz. to the figure 4 and five of the divisions or strokes, which will denote 45, and under that point on the slide is the figure 9, now to be called 90, for the product required.

4th. Multiply 2 by 12.

Here 12 is the multiplier, which point place under i on the middle of the stock, and counting on to the right hand to the figure 2, the multiplicand, under which you have 2, and four of the strokes will give 24 for the product. In order to find 12, the multi-s plier, call the first 1 on the stock for 1, and the middle 1 for 10, and theat two next long strokes will represent. > 12 for the figures sought..

5th. Multiply 30 by 12.

Keep the slide as before, and count 3 for 30 on the stock, under which, on the slide, you will have 3 and six!! strokes, which will represent 360 for the product. If it had been 3 multi-s plied by 12, this point would then have been called 36.

Division.

Division is the reverse of multiplication, and is performed by placing. the figure and division denoting the dividend on the slide under the figure, and division denoting the divisor on the stock, and under unit, or [ 1, on the stock, you have the answerór quotient on the slide.

Examples.

6th. Divide 4 by 2.

Here you place 4, the dividend on the slide under 2, the divisor on the stock, and under 1 on the stock you have 2 on the slide for the quotient required.

7th. Divide 8 by 4.

Keep the slide as before, and look for 8 upon it for the dividend over which you have 4 on the stock for the divisor, then, under 1 you have 2 on the slide, for the quotient of 8 divided by 4.

8th. Divide 30 by 2.

Here you place 3, signifying 30, on the slide under 2 on the stock, and under 1 on the stock you have 15 on the slide for the quotient, by calling 1 for 10, and counting onwards to 15,. which is five of the long strokes to wards 2.

Rule of Three

TW

The rule of thrce, or rule of pro-22 portion, is performed nearly in the ob same manner as multiplication. Fix the second term on the slide underv the first term on the stock, and under the third term on the stock, will bevad

9th. If 2 gives 4 what will 4 give? Place 4, the second term on the slide under 2, the first term on the stock and under 4, the third term on the stock will be 8, the fourth term on the slide for the answer required.

10th. If 3 gives 6 what will 8 give? Keep the slide fixed as before, and you will perceive that 6 the second term is under 3 the first term, and under 8 the third term, you will find 16 by counting 1 on the middle of the slide for 10, and six of the longer strokes for the answer required.

11th. If 24 gives 36, what will 42 give?

Here place 36 on the slide under 24 on the stock, and under 42 on the stock you will have 63 on the slide, for the answer required.

Thus far I have proceeded with the common line of numbers which are laid down upon every sliding-rule, and shall now commence with the description and illustration of the principles and properties of the other side of the rule, which I have denominated the improved sliding-rule.

(To be continued.)

MECHANICS' INSTITUTIONS.

We quote the following excellent remarks from the letter of a correspondent of the Liverpool Mercury. Although they refer immediately to the Liverpool School of Arts, they admit of universal application to Me chanics' Institutions to all Institutions, at least, which are conducted with as pure a regard as this of Liverpool is, for the moral and intellectual improvement of the mechanical classes:

"When the advantages of the Mechanics School of Arts are duly contemplated, and the importance of scientific improvements fairly taken into account, it is astonishing that the young tradesmen and apprentices of our time do not anxiously press forward to an institution so highly calculated to advance their interests and enlarge their prospects. If master tradesmen wish to have their young men well instructed,

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and parents their sons usefully employed in those leisure hours which are too often spent in irregularity and vice, they will enter into a virtuous coalition, and by giving a right direction to the mind and pointing out the benefits of mental cultivation, the School of Arts may be rendered an important blessing to society. Masters and employers will find their advantage in the superior skill and judgment of well-educated apprentices; parents will have the satisfaction of seeing their sons advancing in the path of knowledge, free from the vicious habits attendant upon ignorance and idleness; and the rising race of mechanics will be amply rewarded by the fair prospect of preferment, which art improved by science will undoubtedly command. It is well known that there is no period of life so dangerous to the morals of young persons as that which intervenes between their removal from school and the expiration of their apprenticeship; that is the time in which injurious associations are formed and bad habits contracted. Let the vacant hours of that unguarded season be rationally and usefully employed, and every youth that is thus led on in the path of science, and trained up in the way he should go, will be an honour to himself and a credit to his country. If merchants, master-tradesmen, and artists will do their duty, the School of Arts and all similar institutions may be converted into moral engines of great national importance. What can be more easily effected than to give preference on all occasions to the studious, the moral, the skilful, and the correct? The names of the pupils and their various mechanical engagements are registered on the books of the Institute, Let application be made to the directors for persons of talent and skill, and as soon as it is discovered that merit is rewarded, and that men are selected to places of trust and emolument according to the scientific and moral character they have obtained, journeymen and apprentices will press forward to the School, and a noble strife for preferment and fame will be encouraged by every ingenious and virtuous artisan. In the course of a few weeks, I shall have occasion to apprentice my own son to some useful trade, and such is the high opinion I entertain of the advantages to be derived from the School of Arts, that I will, if possible, introduce a clause into the indenture, making it a part of his duty to become a pupil in that valuable institution,"

DESCRIPTION OF AN IMPROVED PAN

TAGRAPH.

J

L

1

Considering the simplicity of its principle, and its utility in practice, you will not perhaps refuse a corner your useful Magazine to the following account of it. I also subjoin a figure and description of a pantagraph, which I believe is not very generally known, but which, from its portability (packing into less space than a fishing-rod) and the greater surface it is capable of reducing without the trouble of shifting, possesses many superiorities (in the reduction of large plans, &c.) over the instruments in more common use.

The pantagraph consists of four brass rods, two long and two short, moving on fixed centres E, C, G, O; a sliding-box attached to a cylindric tube moves upon the longer rod EI, and another on the shorter rod OG, both these rods are graduated for the different proportions; a similar tube is fixed at T. One of these tubes contains the tracer, another the pencil, and the third the fulcrum. The position of each varying under different circumstances, as is evident from the following

B

D

Sir,-Your correspondent "Keysoo" is wrong in supposing that "the pantagraph is hardly ever used;" on the contrary, there are few instruments more useful to the practical draughtsman. The encyclopedias, it is true, give but a slight account of its construction, and none whatever of its mathematical principles; even that in Adams' Geometrical Essays, is slight and unsatisfactory.

Let E, C, G, O, in the above figure represent the centres on which the parallelogramic part moves, E the common centre of the rods ET, EI,T the fixed tube, and B and D the two moveable ones. Now, if B be the fulcrum, it is evident that the ratio of the moving points T and D will be, TB:TE::DB:DO, in any position of the line TDB or distance of the points T, D, B, because the triangles TgD, DOB, are similar.. distances TB, DB, are in the same ratio as the