OF THE GEOMETRIC LINES WHICH FOLLOW NEXT, and the uses thereof. And first how by means of these we can increase or decrease in any ratio all areas of figures.

Operation VIII.

he lines that next follow the Arithmetic (explained above) are called the Geometric Lines from their being divided in geometrical progression out to 50. From these we gather various uses; and first, they serve us for finding the side of a plane figure that has a given ratio to another [similar] that is given. For example, given the triangle ABC, we wish to find the side of another triangle that has the [area I ratio to it of 3:2. Select two numbers in the given ratio; let these be for instance 12 and 8. Taking line BC with a compass and opening the Instrument, fit this to points 8-8 of the Geometric Lines; then, without changing the opening, take the distance between points 12-12. If we now make a line of that length the side of a triangle, corresponding to line BC, the surface will doubtless be three-halves that of triangle ABC.

     The same is to be understood of any other kind of figure, and likewise we can do the same with circles, making use of their diameters or radii as of the sides of rectilineal figures. And let people with little schooling note that the present operation shows the increase or diminution of all plane surfaces; as for example having a map that contains, say, 10 campi of land, we want to draw one that contains 34. Take any line of the ten-campi map and apply this crosswise to Points 10-10 of the Geometric Lines; then, without again moving the Instrument, take the crosswise distance between points 34-34 of the same Lines; upon such a length describe your map similarly to the first, according to the rule taught above in Operation 3, and you will have the desired map containing exactly 34 campi.

HOW WITH THE SAME LINES WE CAN FIND THE ratio of two similar plane figures.

Operation IX.

et there be given, for example, two squares, A and B, or indeed any two other figures of which those two lines A and B designate homologous sides. We want to find what ratio there is between the areas. Take line B with a compass and, opening the Instrument, fit this to any pair of points of the Geometric Lines, say to points 20-20. Then, not altering the Instrument, take line A with the compass and see what number this fits when applied to the Geometric lines; finding it to fit, say, at number 10, you may say that the ratio of the two areas is that which 20 has to 10; that is, double. And if the length of this line does not fit exactly at any of the graduations, we must repeat the operation and, trying other points than 20-20, get both lines exactly fitted at some [marked] points, when consequently we shall know the ratio of the two given figures, that being always the same as that of the two numbers of the two points at which the said lines fit for the same opening of the Instrument. And given the area of one of two maps you will find the area of the other in this same way, as for example: The map with line B being 30 campi, how large is map A? Fit line B crosswise to points 30-30 and then see to what number line A fits crosswise; that many campi you shall say are contained in the map with line A.

HOW YOU CAN CONSTRUCT A PLANE FIGURE similar and equal to several others given.

Operation X.

et there be given for example three similar figures of which lines A, B, and C are homologous sides; we are to find a single figure similar to these and equal to all three. Take with a compass the length of line C, and opening the Instrument fit this to whatever number along the Geometric Lines you wish. Say it was applied to points 12-12; then, leaving the Instrument in that position, take line B and see what number it fits on the same Lines. Let this be for example at 9-9, and since the other fitted at 12-12, add together the two numbers 9 and 12, keeping in mind 21. Then take the third line, A, and in the same way see what number on the same Lines it fits crosswise; and finding it to fit, say, at 6-6, add 6 to the 21 you were keeping in mind, for 27 in all. Then take the crosswise distance between points 27-27 and you will have line D, upon which you construct a figure similar to the other three that were given, and this will be of a magnitude equal to all those three together. In the same way we may reduce to one any number [of figures] that are given, provided that they are all similar to one another.

GIVEN TWO UNEQUAL SIMILAR FIGURES, TO find a third, similar to these and equal to their difference.

Operation XI.

he present operation is the converse of that already explained in the preceding chapter,' and it is carried out in this way. Given, for example, two unequal circles of which the diameter of the greater is line AA and that of the smaller is line BB, to find the radius of the circle that is equal [in area] to the difference between circles A and B. Take with a compass the length of line A (the larger), and opening the Instrument apply this length to any point you like in the Geometric Lines; let it be fitted, for example, to the number 20-20. Without altering the Instrument, see at what point of the same Lines you can fit line B. Finding this to fit, say, at number 8-8, subtract this from 20 and get 12. Taking the distance between points 12-12 you have line C, [the diameter] whose circle will equal the difference between the two circles A and B. What is thus exemplified in circles by way of their radii' is to be understood to be the same when we operate with one of their homologous sides.

SQUARE ROOT EXTRACTION WITH THE HELP of these same lines.

Operation XII.

n the present chapter three different ways of proceeding in the extraction of square root will be explained: one for numbers of medium size, one for large numbers, and the third for small numbers, meaning by "numbers of medium size" those in the region of 5,000, by "large" those around 50,000, and by "small" those around 100. We shall begin with medium-sized numbers first. To find and extract the square root of a given medium number, then, the Instrument must first be set. This is done by fitting crosswise to 16-16 on the Geometric Lines, the distance of 40 graduations taken lengthwise along the Arithmetic Lines. Then take away from the given number its last two digits, which denote the units and tens, the number thus left being taken crosswise on the Geometric Lines and measured lengthwise along the Arithmetic; what is found will be the square root of the given number. For example, you wish to find the square root of 4,630. Take away the last two digits (the 30) and 46 remains; therefore take 46 crosswise on the Geometric Lines and measure this lengthwise along the Arithmetic. There you will find it to contain 68 graduations, which is the approximate square root sought.
      Two things are, however, to be noted in using this rule. The first is that when the last pair of digits (taken away) exceeds 50, you should add a unit to the number that remains. Thus if, for instance, you want to take the root of 4,192, then since 92 exceeds 50 you should use 42 instead of the 41 that remained; for the rest, follow the above rule.
      The other caution to be noted is that when what remains after removing the last two digits is itself greater than 50, then since the Geometric Lines do not go beyond 50 you must take the half, or some other I-aliquot] part of the remaining number, and using this distance you must geometrically double or multiply the number [obtained] according to the part taken;" the final distance, thus multiplied, when measured lengthwise along the Arithmetic Lines, will give you the root you sought. For example we want the root of 8,412. The Instrument having been set as above, and the last two digits being removed, there remains 84, a number not on the Geometric Lines. So you take its half, or 42, and then having taken crosswise the distance between points 42-42, you will have to double this [distance] geometrically. This can be done by widening the Instrument until the said distance fits to some number of the Geometric Lines for which a double exists [on those Lines], as for example would be done by fitting it to 20-20 and then taking the distance between points 40-40. This, measured finally [lengthwise] along the Arithmetic Lines will show you approximately 91 2/3 which is about the root of the given number 8,412. And if you had been obliged by the given number to take one-third, to triple that geometrically you fit it crosswise to a number of the Geometric Lines for which there is a triple, as it would be for 10 to take 30, or for 12 to take 36.
      As to the manner of dealing with large numbers, that differs from the above only in the setting of the Instrument and in your taking away the last three digits from the given number. To set the Instrument take 100 lengthwise along the Arithmetic Lines and fit this crosswise to points 10-10 on the Geometric. This done, to get the square root of 32,140, for example, take away the last three digits, leaving 32, and take this crosswise on the Geometric Lines; measuring this then along the Arithmetic, you have 179, the approximate root of 32,140. The same caution noted in the preceding operation should be strictly observed in this, for when the three digits taken away exceed 500 you must add a unit to the remaining number, and if the latter exceeds 50 you must take a part of it (half, one-third, etc.), geometrically doubling or tripling what you get for the part taken, in the manner explained above.
      For small numbers set the Instrument in the first way, putting 40 against 16-16, and then take the given number crosswise on the Geometric Lines, without having taken away any digits. Measuring this distance lengthwise along the Arithmetic Lines" you will find the desired root as a whole number with fraction. Note that here the tens of the Arithmetic Lines must serve you as units, and the units as tenths of a unit. For example, we want the root of 30. Set the Instrument as said by placing 40 taken lengthwise along the Arithmetic Lines at 16-16 on the Geometric, from which take crosswise the distance between points 30-30 and measure this lengthwise along the Arithmetic. You will find 55 graduations, which here means five units and five tenths (that is, 5 1/2), which is the approximate root of 30. Note that in this rule you should again observe the instructions and cautions taught for the other two rules.

RULE FOR ARRANGING ARMIES WITH UNEQUAL fronts and flanks.

Operation XIII.

o arrange the front line equal to the flank, it obviously suffices to take the square root of the given number of soldiers. But when we wish to arrange an army having a given number of soldiers so that front line and flank are unequal and in a given ratio, then it is necessary to proceed differently in resolving the problem, as explained in the following example.
      Let it therefore be given to us to find the front and flank of 4,335 soldiers arranged in such a manner that for every five forming the front there shall be three along the flank. To carry this out with the help of our Instrument, consider first the numbers of the assigned ratio, 5 to 3; add a zero to each, pretending they mean 50 to 30. To find the front, we shall take 50 lengthwise along the Arithmetic Lines, and using a compass, fit this distance crosswise to the Geometric Lines at the number obtained by multiplying together the numbers of the given ratio (which in the present example gives 15); with the Instrument left at that setting, we take crosswise on the Geometric Lines the distance between points marked by a number to be determined as follows. Remove the tens and units from the number of soldiers given, leaving in the present example 43; that distance measured lengthwise along the Arithmetic Lines will give us the front line of such an army, which will be 85 soldiers. The flank will be found in the same way, taking 30 lengthwise along the Arithmetic Lines and setting this crosswise to 15-15 on the Geometric Lines, where we next take crosswise the interval between points 43-43, and this measured lengthwise along the Arithmetic Lines will give us 51 for the flank. The same procedure will hold for any other number of soldiers and for whatever ratio is required of us, noting (as was said of square root) that when the units and tens taken away from the given number exceed 50, a unit is added to the hundreds that remain, etc. Nor do I wish to omit that having found the front line by the rule explained above, you can find the flank by another and speedier means and using only the Arithmetic Lines, working as follows. Having already found 85 for the front in the above example, and the numbers of the ratio being 3 to 5, which is as if we said 50 to 30, or 100 to 60; etc., the 85 taken lengthwise along the Arithmetic Lines can be fitted crosswise to 100-100 on those same Lines, and at once the interval crosswise can be taken between points 60-60 on them; this, measured lengthwise, will show us the same number, 51, that was found by the other manner of working.
      And this operation explained under the example of armies is to be understood as the rule for one class of algebraic equations-the one for the square of an unknown set equal to a number," whence all the questions solved by that [algebraic rule] are also resolved by operating with our Instrument in the way just explained.

DISCOVERY OF THE MEAN PROPORTIONAL BY means of these same lines.

Operation XIV.

ith the help of these Lines and their divisions we can with great ease find between two lines, or two numbers, the mean proportional line or number, in this manner. Let the two given numbers, or the two measured lines, be for example 36 and 16; take with a compass the length of one, say 36, and opening the Instrument fit that to points 3636 of the Geometric Lines; then without moving the Instrument take the distance between points 16-16 of the same Lines and measure this along the same scale, finding it to be 24 graduations, which is precisely the mean proportional number between 36 and 16. Note that to measure the given lines we may make use not only of the scale marked on the Instrument, but of any other whatever when that of our Instrument is too small for our purposes.
      Note furthermore that when the lines and the numbers that measure them are very large (that is, exceed 50 which is the largest number marked on our Geometric Lines), our intent can nevertheless be carried out by operating with [aliquot] parts of the given numbers, or with other smaller numbers having the same ratio as the original numbers; and the rule is this. Say that we want to find the mean proportional number between 144 and 81, both of which exceed 50. Take 144 lengthwise along the Arithmetic Lines, to be applied crosswise on the Geometric Lines; since there is no number in the latter that large, I shall in imagination take a part of this number 144, which shall be (say) one-third or 48, and then fit the distance so taken crosswise to points 48-48 of the Geometric Lines. Then, thinking of one-third of 81 (the other given number), which is 27, 1 shall take that number crosswise on the same Geometric Lines, and this, measured along the Arithmetic, will give me the mean proportional sought, which is 108.