Square number

“ Three numbers whose squares are in A.P., with a common difference of 7, are 113/120, 337/120, 463/120; and with Pg 164a common difference of 13 are 80929/19380, 106921/19380, and 127729/19380. In the case of whole square numbers the common difference will always be divisible by 24, so it is obvious that our squares must be fractional. Readers should now try to solve the case where the common difference is 23. It is rather a hard nut.
129.—THE BATTLE OF HASTINGS.—solution
Any number (not itself a square number) may be multiplied by a square that will give a product 1 less than another square. ”

“ From this and the last article it appears, that, whether the sides of a rectangle be a whole or a fractional number of inches, the number of square inches in its surface is the product of the numbers of inches in its sides. The square itself is a rectangle whose sides are all equal, and therefore the number of square inches which a square contains is found by multiplying the number of inches in its side by itself. For example, a square whose side is 13 inches in length contains 13 × 13 or 169 square inches. ”

“ MAGIC SQUARE, a square divided into equal squares, like a chess-board, in each of which is placed one of a series of consecutive numbers from 1 up to the square of the number of cells in a side, in such a manner that the sum of the numbers in each row or column and in each diagonal is constant.
Fig. 1. From a very early period these squares engaged the attention of mathematicians, especially such as possessed a love of the marvellous, or sought to win for themselves a superstitious regard. ”

“ We know that
square ( 25.4 ) must be a little larger than square ( 25 ) = 625 so that
it must be 646.0.
To extract a square root, we set the indicator over the number on the A
scale and read the result under the hair-line on the D scale. When we
examine the A scale we see that there are two places where any given
number may be set, so we must have some way of deciding in a given case
which half of the A scale to use. The rule is as follows:
(a) If the number is greater than one. For an odd number of digits to
the left of the decimal point, use the left-hand half of the A scale. ”

“ The sum of the digits is 59, the sum of which is 14, the sum of which is 5 (which I call the "digital root") , and therefore I know that the number cannot be a square, and for this reason. The digital root of successive square numbers from 1 upwards is always 1, 4, 7, or 9, and can never be anything else. In fact, the series, 1, 4, 9, 7, 7, 9, 4, 1, 9, is repeated into infinity. The analogous series for triangular numbers is 1, 3, 6, 1, 6, 3, 1, 9, 9. ”

“ It will be seen that the constant of the adding square is n times that of the subtracting square derived from it, where n is the number of cells in the side of square. And the manner of derivation here is simply to reverse the two diagonals. Both squares are "associated"—a term I have explained in the introductory article to this department.
The third square is a multiplying magic. The constant, 216, is obtained by multiplying together the three numbers in any line. It is "associated" by multiplication, instead of by addition. ”

“ I do not think I have mentioned the bright form of quadrature in which a square is made equal to a circle by making each side equal to a quarter of the circumference. The last squarer of this kind whom I have seen figures in the last number of the Athenæum for 1855: he says the thing is no longer a problem, but an axiom. He does not know that the area of the circle is greater than that of any other figure of the same circuit. ”

“ A square is the whole amount of the root multiplied by itself.
A simple number is any number which may be pronounced without reference to root or square.
A number belonging to one of these three classes may be equal to a number of another class; you may say, for instance, “squares are equal to roots,” or “squares are equal to numbers,” or “roots are equal to numbers.” [2]
(4) Of the case in which squares are equal to roots, this is an example, “A square is equal to five roots of the same;” ”

“ This we will Pg 175do in Diagram 4. The circle K has one-quarter the area of the circle containing Yin and Yan, because its diameter is just one-half the length. Also L in Diagram 3 is, we know, one-quarter the area. It is therefore evident that G is exactly equal to H, and therefore half G is equal to half H. So that what F loses from L it gains from K, and F must be half of Yin or Yan.
159.—THE SQUARE OF VENEER.—solution
Any square number may be expressed as the sum of two squares in an infinite number of different ways. ”

“ The given number must not itself be a square, because a square multiplied by a square produces a square, and no square plus 1 can be a square. My remarks throughout must be understood to apply to whole numbers, because fractional soldiers are not of much use in war. ”

“ With the exception of the top squares, every square is divided into two parts by a diagonal, the units being written on one side and the tens on the other, so that when a multiple consists of two figures they are separated by the diagonal. Fig. 1 shows the slips corresponding to the numbers 2, 0, 8, 5 placed side by side in contact with one another, and next to them is placed another slip containing, in squares without diagonals, the first nine digits. The slips thus placed in contact give the multiples of the number 2085, the digits in each parallelogram being added together ”

“ Here the directions of the railways will resemble the moves of a rook in chess—that is, we may move in any direction parallel to the sides of the diagram, but not diagonally. Therefore the first town (or square) visited must be a black one; the second must be a white; the third must be a black; and so on. Every odd square visited will thus be black and every even one white. Now, we have 23 squares to visit (an odd number) , so the last square visited must be black. But Z happens to be white, so the puzzle would seem to be impossible of solution. ”

“ A remarkable feature is the bare statement of a number of very close approximations to the square roots of numbers which are not complete squares. Others occur in the Metrica where also a method of finding such approximate square, and even approximate cube, roots is shown. Hero’s expressions for the areas of regular polygons of from 5 to 12 sides in terms of the squares of the sides show interesting approximations to the values of trigonometrical ratios. ”

“ The black squares represent the square of the body of the type, one of the units of measurement, and is called an em, the letter M being exactly square. The following is a list of the number of lines to the foot of the respective bodies as made in actual metal types:
A 'font' of type is an indefinite quantity having all the proper proportions of 'sorts,' including capitals and small capitals, lower-case, spaces, points and references, figures, accents, hyphens, ligatures () , &c. ”

“ Square patterns require great care in working them. First saw the wood perfectly square; then, when on the lathe, take the T square; hold the flat edge firm on the bed of the lathe, and the handle against one side of the wood, till both are even, then fix the wheel of the eccentric chuck to that number. Having with the cutter worked one row of patterns, move the chuck 30 numbers, which, if [Pg 124] the wood is quite square, will enable you to do the second row ”

“ And this figurate number is but the shadow of a Geometricall figure, and doth not indeede expresse it fully, neither is such a quadrate rationall: Yet notwithstanding the numerall side of the greatest square in such like number may be found: As in 148. The greatest quadrate continued is 144 and the side is 12. And there doe remaine 4. Therefore of such kinde of number, which is not a quadrate, there is no true or exact side: Neither shall there ever be found any so neare unto the true one; but there may still be one found more neare the truth. ”

“ With the hundred a square is connected. If you subtract from this hundred and square the two squares negative connected with fifty, then one square disappears by reason of the other, and the remainder is a hundred and fifty, minus a square, and minus ten roots. ”

“ Now may it appeare, to your eye, that the great square made of the whole line A.B, and of one of his partes that is A.C, [f.ii.v] (which is equall with D.G.) is equal to two partiall squares, whereof the one is made of the saide greatter portion A.C, in as muche as not only D.G, beynge one of his sides, but also D.H. beinge the other side, are eche of them equall to A.C. The second square is H.E.F.K, in which the one side H.E, is equal to C.B, being the lesser parte of the line, A.B, and E.F. is equall to A.C. which is the greater parte of the same line. ”

“ If you will look at the lettered square you will understand that there are only ten really differently placed squares on a chessboard—those enclosed by a dark line—all the others are mere reversals or reflections. For example, every A is a corner square, and every J a central square. Consequently, as the solution shown has a turning-point at the enclosed D square, we can obtain a solution starting from and ending at any square marked D—by just turning the board about. ”

“ This gives for inside measurements blade, 221⁄2 inches and tongue 16 inches.
I have described the square as the embodiment of a right angle. If the square is not a 36 right angle, or to use common terms, if the tool is “out of square,” that is, if it is in the least inaccurate, its usefulness is destroyed. When the square is inaccurate instead of solving intricate geometrical problems correctly it becomes a snare and a delusion, leading to false results and misfits in general. It is somewhat remarkable how few workmen test their squares. ”

“ All my readers know what a magic square is. The numbers 1 to 9 can be arranged in a square of nine cells, so that all the columns and rows and each of the diagonals will add up 15. It is quite easy; and there is only one way of doing it, for we do not count as different the arrangements obtained by merely turning round the square and reflecting it in a mirror. Now if we wish to make a magic square of the 16 numbers, 1 to 16, there are just 880 different ways of doing it, again not counting reversals and reflections. ”

“ It will be seen that these squares adapt themselves to other work as well as to framing, a quality very few of the combination squares possess, and while combination squares have their special uses and should be in the tool chest of 20 every expert workman, the square pure and simple, like this of Nicholls or similar ones, should never be absent from the “kit” of the ordinary workman, for with it, if he thoroughly understands it, he can accomplish all that is possible even with a combination square. ”

“ The definition of a square assumes more than is necessary. For if a four-sided figure have all its sides equal and one angle a right angle, it may be shewn that all its angles are right angles; or if a four-sided figure have all its angles equal, it may be shewn that they are all right angles.
Postulates. The postulates state what processes we assume that we can effect, namely, that we can draw a straight line between two given points, that we can produce a straight line to any length, and that we can describe a circle from a given centre with a given distance as radius. ”

“ Suppose one of the parts to be thing, and the other ten minus thing. You multiply thing by itself, it is a square; then by two and seven-ninths, this makes it two squares and seven-ninths of a square. You afterwards multiply ten by ten; it is a hundred, which much be equal to two squares and seven-ninths of a square. Reduce it to one square, through division by nine twenty-fifths ”

“ A square unit is the square described on a line whose length is the linear unit. Thus a square inch is the square described on a line whose length is an inch; a square foot is the square described on a line whose length is a foot, &c.
3. If we take a linear foot, describe a square on it, divide two adjacent sides each into twelve equal parts, and draw parallels to the sides, we evidently divide the square foot into square inches; and as there will manifestly be 12 rectangular parallelograms, each containing 12 square inches, the square foot contains 144 square inches. ”

“ Thus we are more apt to-day to omit the oblong and rhomboid as unnecessary, and to define "rhombus" in such a manner as to include a square. We use "parallelogram" to cover "rhomboid," "rhombus," "oblong," and "square." For "oblong" we use "rectangle," letting it include square. Euclid's definition of "square" [Pg 149] illustrates his freedom in stating more attributes than are necessary, in order to make sure that the concept is clear; for he might have said that it "is that which is equilateral and has one right angle." We may profit by his method, sacrificing logic to educational necessity. ”

“ This is because one fifth of the diameter fails to be represented four times in the circle's circumference. For example: if we multiply the perimeter of a square by one-fourth of any line one-fifth greater than one side, we can in like manner make the square's area to appear one-fifth greater than the fact, as is done by taking the diameter for the linear unit instead of the quadrant of the circle's circumference. ”