OF THE STEREOMETRIC LINES, AND FIRST, how by means of these all similar solids can be increased or diminished in any given ratio.

Operation XV.

he present Stereometric Lines are so called because their divisions are according to the ratios of solid bodies, out to 148; from them we shall collect many uses. The first of these will be that proposed above; that is, given one side of any solid body how we may find the side of another [similar one] that has a given [volume] ratio to it. For example, let line A be the diameter of a sphere, or to speak more familiarly a ball, or the side of a cube or other solid, and let it be required to find the diameter or side of another similar solid having to it the ratio which 20 has to 36. Take line A with the compass, and opening the Instrument fit that to points 36-36 of the Stereometric Lines; this done, take next the distance between points 20-20, which will give line B, the diameter (or side) of the solid which is to the other, whose side is A, in the given ratio of 20 to 36.

GIVEN TWO SIMILAR SOLIDS, TO FIND THE Ratio between them.

Operation XVI.

he present operation is not much different from the one explained above, and can be solved with great ease. If therefore, we are given the two lines A and B, [sides of similar solids], and are asked what ratio exists between their similar solids, we shall take one of these with the compass, for example A, which we shall apply (opening the Instrument) to some number on the Stereometric Lines. Let it fit, say, at 50-50; taking next the length of the other line, B, see at what number it can be fitted, and it being found (for example) to fit at 21-21, we shall say that solid A has to solid B the ratio of 50 to 21.

GIVEN ANY NUMBER OF SIMILAR SOLIDS, TO find a single one equal to all of them together.

Operation XVII.

iven the three lines A, B, and C, sides of three similar solids, we want to find one equal to all these. To do this, take with a compass line A, which is applied to some point on the Stereometric Lines; let this be for example to points 3030. Not moving the Instrument, consider the number at which line B fits; and finding it to fit, for example, at 12, add that number to the number 30 already I mentioned, getting 42, to be held in mind. Then taking line C with the compass, consider the number along the Stereometric Lines at which this fits, and let this be at 6-6, for example. Adding this number to the other, 42, we shall have 48, so that taking C the distance between points 48-48 the line D will be found, whose [similar] solid will equal all three given ones, A, B, and C.

EXTRACTION OF CUBE ROOT

Operation XVIII.

e shall explain two different ways for the investigation of the cube root of any number. The first will serve us for medium numbers, and the second for large ones, meaning by "medium numbers" those not exceeding 148 after we have removed its units, tens, and hundreds. To extract the cube root of these, the Instrument will first be set by fitting crosswise to points 64-64 of the Stereometric Lines the distance 40 taken lengthwise along the Arithmetic Lines. That done, remove the final three digits from the given number and take what remains crosswise on the Stereometric Lines; and measuring this then lengthwise on the Arithmetic Lines, what is found will be the cube root of the given number. For example we seek the cube root of 80,216; the Instrument having been set as above, and the final three digits being removed, 80 remains; therefore take 80 crosswise on the Stereometric Lines and measure this lengthwise along the Arithmetic. You will find 43, which is the approximate cube root of the given number. Note that if, when the last three digits are removed, more than 148 remains, that being the largest number on the Stereometric Lines, you must then operate by parts. For example, seek the cube root of 185,840. Upon removing the final three digits (840), 186 remains-I say 186 although 185 was left, because here the hundreds digit removed was greater than 5, meaning more than half a thousand, whence to seek a better round thousand I add one more unit so that 186 remains-and since this exceeds 148, let us take its half, which is 93, crosswise on the Stereometric Lines (already set). This distance must then be stereometrically doubled," so fit it crosswise to some number of the Stereometric Lines which has a double [thereon], and the latter being taken crosswise and then measured along the Arithmetic scale will be the cube root sought. Staying with the above example, let us fit the distance between points 93-93 (already taken) to points 40-40 on the Stereometric Lines; then take the distance between 80-80 and measure this along the Arithmetic. This shows us 57, which is the approximate cube root of the given number.
      The other way of operating (for large numbers) will be to set the Instrument by fitting the distance of 100 graduations taken lengthwise along the Arithmetic Lines to 100-100 crosswise on the Stereometric. Let it be so set; then from the given number remove the last four digits and take the remaining number crosswise on the Stereometric Lines, measuring this then lengthwise along the Arithmetic. For example, the number 1,404,988 being proposed, and the Instrument having been set in the above way, remove the final four digits, leaving 140. Take that number crosswise on the Stereometric Lines and measure this lengthwise along the Arithmetic, giving us 112, the approximate cube root of the given number. Do not forget that when the remaining digits exceed 148, the largest number on our Lines, one must operate by parts, as in the other rule explained above.

DISCOVERY OF TWO MEAN PROPORTIONALS.

Operation XIX.

hen there shall be proposed to us twonumbers or two measured lines between which we are to find two others that are the mean proportionals, we shall be able to do this easily by means of the present Lines as will be clear from the following example. Given the two lines A and D, of which one is for example 108 and the other is 32, take the larger with a compass and fit it, opening the Instrument, to the numbers 108-108; then take the distance between points 3232, which will give the length of the second line, B; and measuring this by the same scale with which the given lines were measured, 72 will be found. Then to find the third line, C, fit anew points 108-108 on the same Stereometric Lines, this time to the length of B, once again finding the distance between points 32-32; this will now be the magnitude of the third line, C, and measured on the same scale as before, it will be found to give 48 graduations. Notice that it is not necessary to take first the larger rather than the smaller line, but operating with either in the same way you will always find the same [result].

HOW EVERY PARALLELOPIPED CAN BE reduced to a Cube by means of the Stereometric lines.

Operation XX.

et there be given the parallelopiped solid whose dimensions are unequal, as 72, 32, and 84; what is sought is the side of the cube equal to it. Take the mean proportional between 72 and 32 in the way explained in Operation 14-that is, take 72 lengthwise along the Arithmetic scale and set this crosswise against 72-72 on the Geometric Lines (although because they do not go that far, you set it against one-half, or 36, and then take next the other number, that is, 32-32, crosswise on the same Lines, or rather you take its half, or 1616, having likewise cut the original 72 in half); what is found will clearly be the mean proportional number between 72 and 32. Then measure this along the Arithmetic Lines and you will find it to be 48, whence you now set this crosswise to the same number, 48-48, of the Stereometric Lines, and without altering the Instrument you take crosswise the third number of the given solid, which was 84. The operation will then be completed, because by making this line [the distance between points 84-84] the side of a cube, that will truly be equal to the given solid. Measuring it along the Arithmetic scale you will find it to be about 57 1/2.