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duct of any two numbers, call the greater factor the sum, and the less the difference of two numbers, which may be found by the before-mentioned rule; and the difference of the square of these two numbers, will be the required product. Thus, let the two factors to be multiplied be 6 and 12; then 6+129 the greater number, and

2

12-63 the less; now 92— 3281

2

972, the product required.

Any number multiplied by the reciprocal of any other number, will be the quotient of the said number divided by the other number; and by having the reciprocals of numbers in the form of decimals, they may be annexed to the whole numbers, and be used with them with great ease; and division may be performed in the same manner as multiplication, by taking the reciprocal of the divisor for a factor, instead of taking as a divisor the whole number: thus, let the number 15 be to be divided by 5, whose reciprocal is.2, then 15.2

=7.6, and 15-.27.4, and 7.62.

2

2

7.4257.76-54.763 the quotient; the same thing may be more easily performed by using the decimal as a whole number, and cutting off a proper number of figures and decimals at the last, which may also be done in multiplying whole

numbers with decimals.

From what has been said it will easily appear, that by having a table of squares and reciprocals of all the numbers between 1 and 10,000, that the product or quotient arising from any two numbers under that sum may be found by the simple operation of taking the difference of the squares of the half sum, and difference of the numbers to be multiplied and divided; and the easy method of constructing such a table, and of finding the square of any number by it, that is not included in the table, as also, in correcting the squares in case of error, must render this method of computation very easy. These things can be done much readier than in logarithms, and a table of squares may be carried to a much greater extent than a table of logarithms with much less trouble; but a table of squares of all numbers from 1 to 10,000 may be sufficient for most purposes.

Mixed numbers or fractions may be reduced to decimals, and then be worked as whole numbers, cutting off a proper number of figures for decimals, at the end of the operation; and it must be observed, that there will be always double the num

ber of decimals in the square of any number, as there are in that number.

The nature of squares farther explained by the Parabolic Curve.

The equation of the common parabola is axy2, and when the invariable quantity a is equal to 1, then the equation is x=y2. From this it appears that the abscissa of the curve is every where equal to the square of the corresponding ordinate; hence, a table of squares would be no other than parabolic abscissas, and their corresponding ordinates are the square roots, or numbers of which they are squares.

tioned property of numbers, it appears, By the application of the before-menthat in any parabola whose equation is xy, the difference of any two abscissas, is equal to the product, or rectangle, of the sum and difference of their corresponding ordinates. Let y any ordinate and y will be the corresponding abscissa; and let y+a be another ordinate, then the corresponding abscissa will be y2+ 2 ay+a2, and their difference is y2+2 ay +'a2—y3 =2 aya2; the sum and difference of the ordinates are 2y+a and a; hence, their product is 2 ay + a2.

If a parabola of the kind above-mentioned were properly divided, calculations might be made with it after the manner of the logarithmic scale.

Of the method of increasing the squares in

case of Fractions or great numbers. When a table of squares is made, the square of any number is readily found by as far as it extends; but sometimes fractions, which should be converted to decimals, enter into the operations, and then it will be necessary to increase the squares so as to answer our design, as also in case the square of a greater number should be required. The square of any number may be readily found by multiplying the number by itself; but when we have a table of squares, this operation may be performed with much less trouble.

If the side of a square be increased by a given quantity, the square of the increased line will exceed the former square by twice the rectangle made by that quantity and the side of the first square, together with the square of the aforesaid part.

This appears by squaring any binomial, one part of which is the root or side, and the other the increased quantity; thus a+b2= a2 + 2 a b + b2. If it were

required from the square of 10, which is 100, to find the square of 10.4, or 104; then 10 X.4 X 2 +.16 + 100 = 108.16, the square of 10.4 or 10816 the square of 104; and the square of n+.5=n2+ n+, which might often be applied very usefully.

Of the Construction of the Scale.

Whether these numbers be considered as being too troublesome for general use, I leave to the judgment of others; certainly in multiplication they may be carried to a greater extent than logarithms, though it must be allowed that the operation will be a little more trouble. Division by reciprocals may seem rather tiresome, but a table of reciprocals extending to 1,000 would be sufficient for most purposes; indices might be put, to denote the number of ciphers, and double numbers and reciprocals might be prefixed to the squares instead of single ones, by which the trouble of dividing by 2 would be saved. A scale constructed on the principle of these numbers would be very useful both to mathematicians and navigators, in making calculations; and in many cases, especially in great numbers, it would succeed better than the Gunter. Its construction is as follows:

Draw a straight line of 1 or 2 feet in length, divide it into 10 equal parts, each of which must again be divided into 10 other equal parts; and if the scale is suffi. ciently long, these latter divisions may again be divided into 10 other parts. Call this line b. Draw another line c on the right hand of b, and nearly close to it, which divide into 20 equal parts, each of which must be divided into tenths, &c.; next make two narrow columns on the right-hand side of these lines, the first of which refers to the line b, and contains the numbers 1, 2, 3, &c. to 10, over against the division of the line; the second column refers to the line c, and is numbered 1, 2, 3, &c. to 20.

To the left hand of the line b, and nearly close to it, draw another line a; and suppose the numbers of the line b to be 1,000 2,000, &c. to 10,000, then divide the line a into 100 unequal parts, so that each division may stand over against its square on the line b, and number every tenth division 1, 2, 3, &c. to 10.

To the right hand of c draw another line d, of half the length of the former lines; divide it reciprocally into 10 parts, and each of these parts must again be divided into 10 other parts reciprocally. Number the greater divisions, 10, 9, 8, &c. to 1,

which will be the whole length of the line.

Now, it must be observed that when the numbers of the line a are taken 1, 2, 3, &c., those on b must be taken 10, 20, 30, &c.; and when they are taken 10, 20, 30, &c., on a, those on b must be called 1,000, 2,000, &c.; and if taken 100, 200, &c. then those on b must be taken 100,000, 200,000, &c. &c.

The Use of the Scale.

To find the sum and difference of any two numbers. Take the distance of the lesser number, on the line c in the compasses, and set one foot on the point of the greater in the same line, and extend the other foot upwards for the sum, and downwards for the difference; and over against the sum and difference, stand the half sum and half difference on the line b.

To multiply two numbers, take the distance of the half difference on the line a in your compasses, and set one foot on the point of the half sum on the same line, and extend the other foot downwards, and you have the product over against it on the line b. When one of the factors is small and the other great, it will be the best to take the divisions of the scale, for the lesser factor ten times greater, by which the dif ference of the squares will be increased in a tenfold proportion on the line b; or, the numbers of that line must be taken ten times less.

To divide any number.

Take the reci

procal of the divisor on the line d in the compasses, and find the half sum and half difference of this quantity and the dividend, as before taught; and proceed exactly in the same manner as in multiplication; but when the numbers of the lines b and c are taken 10, 20, 30, &c., and the reciprocals are called 1, 2, 3, &c., they will be 100 times too long; therefore the line b, in finding the quotient, must be taken two ciphers less, that is, instead of calling the numbers 1,000, 2,000, &c., they must be called 10, 20, &c.; and when the numbers of the reciprocal line are taken 10, 20, 30, &c., they will be 10 times too great, and then the line b, instead of being 1,000, 2,000, &c., must be taken 100, 200, &c.: all these particulars will be best understood by considering the theory.

The principal difference between this scale and the Gunter, is, that the lengths of the divisions on the line a increase as the numbers increase, and that the divisions of the line b, on which the products and quotients are found, are of equal length; hence the operation will

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DESCRIPTION OF AMSTERDAM.

WHOEVER is desirous of seeing human ingenuity and human industry most successfully and most extensively exerted, for the purpose of counteracting the injurious effect of one of the most powerful and destructive elements, and by means the most simple, must visit Holland, and more particularly Amsterdam. He will there see and admire the simple and effectual means that have been adopted for the security of the town, by bringing the waters under complete control. The whole extent of the seafront, with the quays and the shipping, is protected from injury by a double stockade of strong, square, wooden posts, known by the name of boomen or barriers, extending at a distance from the quay along the whole line of the city, from the north-west to the south-east corner, a distance of two miles and a half. These large beams of wood are firmly fixed in pairs, with openings between each tier, at certain distances, to allow ships to pass them to and from the quays. Of these openings or passages, there are twenty-one, all of which are closed by night: so that nothing can arrive at, or depart from, the quay, till they are set open. By means of these barriers, the injurious effects of the waves on the wharf wall, by being divided and dispersed, as well as of masses of ice driven down from the northward, are completely obviated. All the quays, and indeed every house in Amsterdam, are built upon piles, and as each of these is a large tree or balk of timber, of forty or fifty feet in length, some idea may be formed of the expense of building in Amsterdam, as well as of the immense quantity of timber that must have been brought thither for this purpose alone. It is recorded, that the number of piles on which the old Town House, now the Royal Palace, is built, amounts to upwards of thirteen thousand. Indeed, the industry of the Dutch is not to be surpassed; and it is exercised with great skill and ingenuity, and also with indefatigable perseverance; otherwise they could never have succeeded in accomplishing such great undertakings with such small means.

On no occasion, perhaps, is this ingenuity and perseverance more displayed than in the means employed in conquering the

waters of the ocean, and in bringing under subjection the rivers, lakes, and canals, with which they are surrounded on either side, by means of sluices, drains, ditches, and windmills; of the last of which, for this and other purposes, such as sawing wood, grinding corn, and crushing seeds for oil, the number in the vicinity of all their towns and cities is perfectly astonishing. These windmills are remarkable objects on the Boulevards of Amsterdam. There are no less than thirty bastions in the line of fortification on the land side, and on each bastion is a windmill, of a description larger than common, for grinding corn, and other purposes. It is whimsical enough that, surrounded as they are with water on every side, there is not a watermill in the whole country. It suited their purpose better to raise a contention between the elements, by employing the wind to drive out the water. Necessity indeed taught the Hollander this: for if it were not for the complete subjection in which the waters are held by this and other means, the city of Amsterdam might at any one moment be altogether submerged. The idea of such a calamity, happening to a city which is stated to contain near two hundred thousand inhabitants, calls for every precaution that can be put in practice to avert it. Of this number of inhabitants, consisting chiefly of Calvinists, Catholics, Lutherans, and Jews, by far the greater part are engaged in some kind of commerce another; few of them in manufactures, except such as are in every-day use, and for home consumption.

or

Many of the artisans and the poorer classes, inhabit the cellars under the houses of the more opulent, and a great many reside constantly on the water, in comfortable apartments built on their trading vessels, more particularly those employed in the inland navigations. In this and many other respects, the Dutch bear a strong resemblance to the Chinese; like this industrious and economical race, they keep their hogs, their ducks, and other domestic animals constantly on board. Their apartments are kept in a state of great neatness; the women employ themselves in all the domestic offices and are assiduous in embellishing their little sitting rooms with the labours of the needle; and many of them have little gardens of tulips, hyacinths, anemonies, and various other flowers. Some of these vessels are of great length, but generally narrow, suitable to the canals and sluices of towns. Each vessel is generally navigated by the members of one family, of which the female part is by no means the least useful, nothing being more common

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