bars prepared for grinding; let us then sup. pose the edge of A to be slightly convex, and that of B slightly concave, or nearly straight, then by grinding A and B together the two edges will meet, but will not be straight, because the convex bar A has ground the lower bar B more concave, and although the two edges come in close contact, yet the form is unchanged, and, however long the grinding should be continued, the object could never be attained. But if we now take a third bar C, the edge of which may be either concave or convex; if concave, and we grind A and C together, the edges of B and C will then be similar, and if placed against one another, the difference will be doubled, and can readily be perceived; these two are then to be ground together, and thus the three edges being alternately and reciprocally ground together, they will mutually cut down and destroy each other's imperfections, and a perfect straight edge will ultimately be produced on all the three. The same theory applies to the levelling of air-pump plates, and other flat surfaces in machinery where great nicety is required, and the best method of producing them is to proceed in the manner above described. [Young Mechanic.] THE WALTHAMSTOW WOODEN SPOON MA. KER. I lately happened to meet with a very old and intelligent individual, in the neigh. borhood of Walthamstow, who was represented to me by his daughter as being no scholar, and who found it convenient to em. ploy his evenings in making one single article-wooden spoons. Every evening in the year, when the evenings were long, he sat down, and with very few tools-a couple of gouges, a plain chisel, a bad file, and two or three knives-he sat down and constructed four spoons every evening, which he formed in a somewhat beautiful manner. You see to what a degree of perfection a mono-mechanic a man who makes but one thingcan attain by directing his attention simply to the making of spoons. The old artist constructed them of different kinds of wood, all of our own country-some of the plumtree, the barberry, (which in day-light is very beautiful,) and the pear-tree, and the cherrytree. He makes four spoons every evening, which he sells at three pence each; and is visited every evening by many persons in consequence of these productions: (they are exceedingly convenient little spoons for tea. caddies, sugar-basins, and so on,) and he feels himself that it is better to make a spoon than to be idle, and that, if he were not sitting still making spoons, he might be sitting somewhere making mischief. He is besides a man of considerable natural taste: he is an entomologist, a butterfly-catcher-but he understands nothing more about them than their commmon names; yet he arranges them very beautifully: he has a very pretty cabinet of butterflies, and moths, and insects, and in the day-time is engaged in that pur. suit. He entertains himself likewise in a garden, enjoys delightful health and spirits, and is a very interesting picture of a happy old man.-[Dr. Birkbeck.] BLACKING.-"A Subscriber" asks for a receipt for making blacking, "as his family is numerous, and purchasing at 18d. per bot. tle is more than he can well afford." We subjoin one which is given in the Annales de Chimie, by the celebrated French chemist Braconot, who pronounces it to be, as com pared with all the other known blackings, from Day's to Hunt's, "undoubtedly the cheapest and the best." Macerate one pound of malt in boiling water till every thing soluble is taken up, add 24 lbs. of plaster of paris well sifted, and 7 ozs. of lamp-black; then evaporate to the consistence of paste; and finally mix up with 1 lb, 2 ozs. of olive oil. It is said to spread very evenly, dry speedily, and shine brilliantly, with very little brushing; while it neither burns nor injures the leather. [London M. Mag.] STEAM BOILER DEPOSITS.-Institution of Civil Engineers, April 3.-In the case of the saline deposition, which accumulates in boil. ers during sea-voyages, it was mentioned as the usual practice merely to blow off a por. tion of water from the boiler, according as it becomes saturated. In short voyages of three or four days, this is found sufficient for the purpose; but for vessels crossing the Atlantic, or on other long voyages, a more efficient plan has been resorted to, by attach ing an apparatus to the engine which pumps out brine from the bottom of the boiler, at the same time throwing in a quantity of clear water equal to what is abstracted. The degree of saturation is indicated by means of an attached thermometer: 218 degrees Fahr. being the boiling point of clean sea water in a steam engine boiler, a range is allowed from that to 227 degrees, which marks the limit of saturation admissible for a steam boiler to be worked with safety.-[Ath.] A Machine uniting all the Mechanical Powers.—Decimal Fractions. 825 A MACHINE IN Which all the MECHANICAL POWERS ARE UNITED.-The preceding figure represents a machine in which all the simple mechanical powers are combined. It consists of a frame A B C D, fastened upon the stand O o by the nut o, and kept to. gether by the pillars V W and B q. The piece E F is first fitted to the frame, having vanes, E F, which may be either moved by the wind, or by a cord fastened at F. This part represents the lever, whose fulcrum is G. A perpendicular axis G A is joined to this lever, and carries the endless screw H, which may be considered as a wedge. This endless screw works in the teeth of the wheel K, which is the wheel and axle; and when K is turned round, it winds upon the axle. I L the cord L M, which, passing round the tackle of pulleys M N, raises the weight P. In order to include the inclined plane in this combination, we must add the plane RQrq, and make it rest on the ground at Q R, and on the pillar B q at qr. When the weight P is placed on this plane, the power will be farther increased in the ratio of QT to TS. The power gained by this combination will be found, by comparing the space described by the point F with the height through which the weight rises in any determinate number of revolutions of F. DECIMAL FRACTIONS.-We now proceed to explain the species of fractions which are called decimal, a word derived from the Latin decem, ten. In doing this it will be neces sary to enter upon the decimal system gen. erally, and to point out the features which distinguish our arithmetic from that of an. cient times. The Greeks and Romans reck. oned as we do, by tens; that is to say, hav. ing given names to the first ten numbers, they made these names serve to reckon all numbers as far as ten tens, or one hundred, for which a new name was introduced; with this they proceeded as far as ten hundreds, or one thousand, where again a new name was adopted. In the symbols by which they represented numbers, they were not fortu. nate; and the Roman method especially, which is often used amongst us, is so clumsy as to make it no matter of wonder why that people never cultivated arithmetic with suc cess. Our method came originally from In. dia through the Moors, who brought it into Spain. It enables us to represent all num. bers by means of ten symbols, one denoting nothing, and the rest standing for the first nine numbers. The value of a figure de pends not only upon the number which it represents, when it stands alone, but also upon the place or column in which it is found. Thus, in 2222 yards, the two on the right hand stands for two yards only; the next to it for 2 tens of yards, or twice ten yards, or twenty yards; the next for two tens of tens of yards, or two hundred yards; the next for two tens of hundreds of yards, or two thousand yards. It is necessary to recall this, which is well known to all our readers, and in which the superiority of the modern system consists, in order to show how simply fractions may be represented by an extension of the same method. In the num. ber 11111, if we proceed from left to right, each unit is the tenth part of the one which preceded it. Thus the first one is ten thousand, the second one thousand, the third one hundred, and so on. The last 1 is simply a unit, which may, introducing fractions, be divided into ten parts, each of which will be one tenth of the unit, and will be represent ed in the common way by If we would carry on the notation just explained, in the case of 11111, we may place one more unit on the right and agree that it shall stand for of the unit. This would give 11111 1, in which the separation is made to avoid confounding this, which is eleven thousand one hundred and eleven yards and one-tenth of a yard, with 111111, which is one hundred and eleven thousand one hundred and eleven yards. In the same way in 11111 1111, the first 1 after the unit's place, or the first which is separated from the rest, stands for onetenth of a yard, the second for one-tenth of a tenth, or one-hundredth of a yard, the third for a tenth of a hundred of a yard or onethousandth of a yard, and the fourth for one. tenth of a thousandth, or one-ten-thousandth part of a yard. Instead of a separation, it is usual to mark a point after the unit's place, and all figures which come before the point are whole yards, pounds, acres, &c., as the case may be, while all which come after the point are fractions of the same. Thus, 12-34 yards stands for 12 yards, 3 tenths of a yard, and 4 hundredths of a yard; 768 stands for 7 tenths, 6 hundredths and 8 thousandths. The cipher is used in the same way as in whole numbers, viz. to keep each number in its proper place. Thus one-hundredth is distinguished from one-tenth by writing the first 01, and the second 1, since the second column on the right of the point is appropriated to hundredths, and the first to tenths. Thus 308 is three tenths and eight thousandths; 0308 is 3 hundredths and 8 ten thousandth parts. These fractions may be represented in another way. Thus, 123, which is one. tenth, 2 hundredths, and 3 thousandths, is also 123 thousandths, or one hundred and twenty-three parts out of a thousand. For if we divide the unit into 1000 parts, one-tenth is 100 of these parts, one hundredth is 10, and two hundredths are 20 of these parts, and three thousandths are three of these parts. Similarly 76 is either 7 tenths and 6 hun. dredths, or 76 hundredths. The rule is: To write a decimal fraction in the common way, let the numerator be the number which fol. lows the point, throwing away cyphers from the beginning if necessary; let the denomi. nator be unity followed by as many ciphers as there are places of figures after the point. By the same rule a number and decimal frac tion may be converted into one common fraction, the numerator being formed by throwing away the decimal point. Thus 7-12 is 7 or 1. The decimal point is always understood as coming after the unit's place, even when there are no fractions. Thus 16 is 16. or 16.000. And any number of ciphers may be placed after a decimal without altering its value. Thus 4 and 40 are the same, the first being 4 parts out of ten, and the second also 4 parts out of ten, or which is the same thing, 40 parts out of one hundred. No frac tion can be converted into a decimal of er. actly the same value, unless its denominator be either 5 or 2, or a product of some num. ber of fives and twos, such as 250, which is the product of 5, 5, 5, and 2. For the changing a common into a decimal fraction is the finding a second fraction, equal in va lue to the first, and whose denominator shall be one of the series of decimal numbers, 10, 100, 1000, &c. There is only one way of altering the terms of a fraction without al tering its value, viz. by multiplying or dividing both numerator and denominator by the same number. It will easily be found by experi ment, and it is proved in books of algebra, that a decimal number, that is, a unit followed by ciphers, is not divisible by any number ex cept it be either 2, 5, or a product of twos and fives. Hence it is impossible that a mul tiplier can be found for 7, for example, which shall make the product a decimal number 2002 Decimal Fractions. 2000 13000. for iso, since the product is always divisible by the multiplicand, there would be a decimal number divisible by 7, which is impossible. Hence there is no decimal fraction exactly equal to, or, or, and so on. Ne vertheless, a decimal fraction can be found as near as we please to any fraction whatever; that is, if we take, and take any fraction as small as we please, for example, Tooooo or '00001, we can find a decimal fraction which shall not differ from by so much as ⚫00001; and if we please, we can come still nearer than that small difference. Suppose it is required to find a decimal frac tion which shall not differ from by so much as Too or 001. Multiply the numerator and denominator by 1000, which gives The numerator 2000, divided by 13, gives the quotient 153 and the remainder 11; so that both 2000 diminished by 11, and 2000 increased by two, are divisible by 13, that s, 1989 and 2002 are divisible by 13, and give the quotients 153 and 154. Of the three fractions 3000 13000 and 1, which have the same denominator, the first is the least, the third is the greatest, and the second lies between the first and third. But the first and third (dividing both numerator and denominator by 13) are 15 and 1 or 153 and 154, and the second is the same as The first and third differ from one another by or 001; hence the second, which lies between them, does not differ by so much as ross from either. We have, therefore, two decimal fractions 153 and 154, the first a little less, and the second a little greater, than, each within roof. The rule derived from this process is-To find a deci. mal fraction which shall not differ from a common fraction by so much as roo, &c. annex as many ciphers to the numerator as there are ciphers in 1000, &c., divide by the denominator, and cut off by the decimal point from the quotient as many places as there were cyphers in 1000, &c., completing the number, if necessary, by adding cyphers to the left, and taking no account of the remainder. Thus 365 is within 1000 of 20. 277777. 1066 1500 The rules for addition, subtraction, multiplication and division, of decimal fractions, are very similar to those in whole numbers. In addition and subtraction the decimal points are to be placed under one another, which will bring units under units, tens under tens, tenths under tenths, &c. The process is then precisely the same as in whole numbers, the decimal point in the result being 238 26656 112 placed under the other points. In multiplication we must proceed to multiply as if there were no decimal points, and after. wards make as many decimal places in the result as were in both the multiplier and multiplicand. For the product of 238 and 112 or 3 and is, by the common rule, 38858 and as one decimal number is multiplied by another by forming a third decimal number, which shall have as many ciphers as both the former ones together, and since the number of ciphers in the de. nominator of a decimal fraction expressed in the common way is the number of places which it will have when the point is substitų. ted for the denominator, the reason of the rule is evident. The product obtained above is 026656 by the rule, one cipher being neces. sary to make up six places. It is moreover evident that is less than 1 or, the latter being 1000000 100000 The rule for division of one decimal by another, as given in many books of arithme. tic, is likely to mislead the student in various cases. From the following principles a rule may be drawn which will apply to every possible case. If there be no decimals ei. ther in the dividend or divisor, the rule has been already explained. Thus the division of 17 by 6 is the same thing as the reduction of to a decimal fraction, since the sixth part of unity repeated 17 times is the sixth part of 17. Again, we must observe that when two fractions have the same denominator, their quotient is the same as the quotient of their numerators. Thus is contained in 7, just as 2 is contained in 17. By the rule, divided by gives, which hav. ing the numerator and denominator both di. visible by 3, is the same as . If then two fractions have the same denominator, the denominator may be rejected in division, and the one numerator divided by the other. Two decimal fractions may be reduced to the same denominator by annexing ciphers to the right of that which has the fewest number of places, so as to make the same number of places in both. For we have shown that a decimal is not altered by annexing ciphers on the right, and we know that two decimals which have the same number of places have the same denomi nator, viz. unity followed by as many ciphers as there are places. If then, we have to divide 42.1 by 0017 we begin by annexing three ciphers to 42.1, which gives 42.1000 and 0017, which having the same denomi. nator, we retain only the numerators, which 21000 403812 are 421000 and 17. It only remains to re- When any decimals are thrown away from a result, it is more correct to increase the last remaining figure by 1, if the first figure thrown away were 5 or upwards. Thus, if out of 13885, we retain only four places, we write it 1389, this being nearer to 13885 than 1388. If we retain 3 places only, we write 139. On the same principle, if we had to mention 278 in round numbers, we should rather call it 300 than 200. BIAS OF THE Mind respectinG THE FUTURE-The common bias of the mind undoubtedly is, (such is the benevolent appoint. ment of Providence,) to think favorably of the future, to overvalue the chances of possible good, and to underrate the risk of possible evil; and in the case of some fortu. nate individuals, this disposition remains after a thousand disappointments. To what this bias of our nature is owing it is not material for us to inquire; the fact is certain, and it is an important one to our happiness. It supports us under the real distresses of life, and cheers and animates all our labors; and although it is sometimes apt to produce, in a weak and indolent mind, those deceitful suggestions of ambition and vanity, which lead us to sacrifice the comforts and duties of the present moment to romantic hopes and expectations, yet it must be acknowledged, when connected with habits of activity, and regulated by a solid judgment, to have a fa vorable effect on the character, by inspiring united (as it commonly is) with pleasing no. Co-OPERATIVE LABORERS.-Many of our readers are no doubt aware that some well. intentioned men have been endeavoring for a long time to effect a great change in society, by establishing a new arrangement, called Co-operation, which assumes that the labor. ers should be at the same time the capital. ists. There can be no sort of objection to this principle, when it is proposed to carry it into action without any prejudice to the existing laws of property; and, no doubt, many of the evils of our social state might be removed, were all persons concerned in the business of production to have a sort of proprietary interest in the commodities pro. duced. The mistake of those who exclu sively call themselves co-operatives, is that of assuming that the love of individual property can be got rid of by a very short pro. cess of reasoning, and neglecting to avail themselves of the many practical modes in which industry might be made more pro ductive than at present, by a union of forces, in which the personal interests of every la. borer would be dependent upon the success of the business in which he is engaged. There are many examples of such real cooperation already existing in the world, some |