DIVISION OF A LINE

FIRST OPERATION

oming to the detailed explanation of the operations of this new Geometric and Military Compass, we shall begin first with the face on which there are marked four pairs of lines with their divisions and scales; among those we shall first speak of the innermost, called the Arithmetic Lines from their division in arithmetical progression; that is, by equal additions which proceed out to the number 250. We shall gather various uses for these Lines; and first: By means of these Lines we can divide any given straight line into as many equal parts as desired, operating in any of the ways set forth below.
      When the given line is of medium size, so that it does not exceed the opening of the Instrument, we take its whole length with an ordinary compass' and apply this distance crosswise, opening the Instrument, to some number land its counterpart on the other arm] on these Arithmetic Lines such that above it, on these same Lines, there is a smaller number contained by the selected number as many times as there are parts into which the given line is to be divided. Then the crosswise distance taken between the points bearing this smaller number will doubtless divide the given line into the required parts.
      For example: Wishing to divide the given line into five equal parts, we take two numbers of which the larger is five times the other, such as 100 and 20, and opening the Instrument we adjust it so that the given distance (taken with the ordinary compass) fits crosswise to the points marked 100 and 100. Then, not again moving the Instrument, take the crosswise distance between the points on the same Lines marked 20 and 20; undoubtedly this will be one-fifth of the given line. And in the same order we shall find any other division, noticing that large numbers are to be taken (but not exceeding 250), because then the operation turns out to be easier and more exact.
      We shall be able to do the same thing operating a different way, and the order will be this. Wishing to divide the line shown below into 11 parts, we shall take one number eleven times the other, as would be 110 and 10, and setting a compass to the whole distance AB we then fit this crosswise (opening the Instrument) to the points 110. Next, being unable to find the distance between points 10-10 along these same Lines, that [region] being occupied by the large hinge, we instead take the distance between points 100-100, narrowing the points of the compass a bit. Then, fixing one of its legs at point B, we mark with the other the point C; the remaining line AC will then be one-eleventh of the whole line AB, and we likewise fix one leg of the compass at A to mark point E near the other end, making EB thus equal to CA. Next, again narrowing the compass a bit, we take the crosswise distance between points 90-90 and apply this from B to D and from A to F, getting two new lines CD and FE which are also one-eleventh of the whole. Transferring with the same order, in either direction, the distances taken between points 80-80, 70-70, etc., we shall find the other divisions, as clearly seen from the diagram.

     Now, when the given line is very short and it is to be divided into many parts, as for example the line AB below which is to be divided into 13 parts, we may proceed by [adapting] this second rule.
      Let the line AB be extended faintly out to C, and let there be marked along this [extension BC] some other lines, as many as you please, each equal to AB; in the present example let there be six of them, so that AC is six times greater than AB .5 It is evident that of the parts of which AB contains 13, all AC will contain 91, wherefore taking all AC with a compass we shall apply this crosswise, opening the Instrument, to points 91-91. Next, narrowing the compass a bit to [fit across] points 90-90, we carry that distance from point C in the direction of A. Marking the point near A, this will give us the 91st part of all CA, which is the 13th part of BA. Then, narrowing the compass bit by bit to 89, 88, 87, etc., we transfer those distances from C towards A, finding and marking the other little parts of the given line AB.

     Now finally, if the line to be divided is very long, so that it much exceeds the widest opening of the Instrument, we can nevertheless take in it the required part, which let be (for example) one-seventh. To find this, we first think of two numbers of which one is 7 times the other, as for instance 140 and 20. Open the Instrument at will and take crosswise the distance between points 140-140; see how many times this is included in the given line. However many times it is contained, that many times is the crosswise distance between points 20-20 to be repeated along the large line, and you will have its one-seventh part if the distance taken between points 140-140 precisely measured the given line. If it did not exactly measure this, it will be necessary to take one-seventh of the excess in the manner previously explained, and by adding that to the distance which was laid off many times along the large line, you will have its oneseventh part to a hair, just as was desired.

HOW IN ANY GIVEN LINE WE CAN TAKE AS MANY parts as may be required.

Operation II.

his operation is the more useful and necessary according as without the help of our Instrument it may be [the more] difficult to find such divisions, which can nevertheless be found instantly with the Instrument. Thus when one is asked to take in a given line some parts, such as (for example) 113 parts out of 197, one just takes the length of the line with a compass and. opens the Instrument so that this length fits crosswise between the points marked 197-197; without again moving the Instrument, take with the same compass the distance between points 113-113, and that will doubtless be the fraction of the line requested, equal to one-hundred-thirteen one-hundred-ninety-sevenths of it.

HOW THESE SAME LINES GIVE US TWO, AND even infinitely many, scales for altering one map into another, larger or smaller.

Operation III.

t is evident that whenever it is required to draw from a given diagram another similar one, large or smaller in any desired ratio, we must use two scales accurately divided, one of which serves to measure the drawing already made, and the other for marking the lines of the drawing to be made. Two such scales we shall always have from the Lines of which we are now speaking. One will be the line already divided lengthwise along the Instrument; this established scale will serve us for measuring the sides of the given drawing. The other, for drawing the new design, must be alterable; that is, it must be capable of increase or diminution at our pleasure, according as the new drawing is to be larger or smaller. This variable scale will be that which we have crosswise from the same Lines by narrowing or widening the opening of our Instrument. For a clearer understanding of the manner of putting these Lines to use, let us take an example. Thus, let diagram ABCDE be given, to which we are to draw another, similar, but based on line FG which is to be homologous (that is, corresponding) to line AB. Here it is evident that two scales must be used, one to measure the lines of diagram ABCDE and another by which will be measured the lines of the diagram to be drawn, and this latter must be greater or less than the other scale according to the ratio of line FG to line AB.

Therefore take with a compass the length AB, putting one leg of the compass at the pivot of the Instrument and the other at any point it strikes, [establishing a measure for the line] which in our example let be point 60. Then take line FG with the compass, and placing one of its legs at point 60, open the Instrument until the other leg falls crosswise exactly on the other corresponding point 60, without again altering the opening of the Instrument. All the other sides of the given diagram will be measured in turn along the lengthwise scale, and immediately the corresponding crosswise distances to these will be taken for the sides of the new diagram. Thus (for example) we wish to find the length of line GH corresponding to BC; take distance BC with the compass and apply this from the pivot of the Instrument lengthwise along the scale, and holding one leg on the point where it falls, say for example at 66, fit the other leg crosswise to the corresponding point 66, according to which measure you will mark line GH, and it will correspond to BC in the same ratio as that of line FG to AB. Notice that when you want to transform a small diagram into one somewhat larger, it will be necessary to use the two scales in the opposite order-that is, use the lengthwise scale for the diagram to be drawn, and the crosswise scale to measure the lines of the given diagram. Thus, for example, let us take the diagram ABCDEF which we wish to transform into another, somewhat larger, and based on line GH which is to correspond to line AB.

To set the scales, take line GH and see how many graduations it contains along the lengthwise scale; seeing this to contain, say, 60, take its corresponding line AB and fit that crosswise to points 60-60, not moving the Instrument again thereafter. Then to find line HI, corresponding to BC, take BC with the compass and find which points it fits along the crosswise scale. Finding this to fit, say, points 46-46, at once take the distance to point 46 lengthwise along the scale and you will have the length of line HI, corresponding to BC. Notice that in this (as in the other) operation it does not suffice to have found the length HI without also finding the point toward which this must be directed so that angle H will be made equal to angle B. Therefore having found [the length of] this line HI, fix one leg of the compass at point H and, with the other, mark faintly a short arc as shown by our dotted line OIN; next the distance between points A and C will be taken, and the number of graduations out to this along the crosswise scale will be sought. This being found to fit, say, at 89-89, take the distance 89 lengthwise with the compass, and holding one leg at G mark the intersection of arc RIQ with the first arc OIN, this being point I toward which you must direct line HI. Then doubtless angle H will equal angle B and line HI will be proportional to BC. In the same way will be found the other points K, L, and M, corresponding to corners D, E, and F.

THE RULE-OF-THREE SOLVED BY MEANS OF A Compass and these same Arithmetic Lines.

Operation IV.

he present Lines are used not only for resolution of various linear problems, but also for some arithmetical rules, among which we place this one corresponding to what Euclid teaches us: Given three numbers, to find the fourth proportional. This is the Golden Rule, called rule-of-three by practitioners who find the fourth number proportional to the three given. To demonstrate the whole thing by an example, we say for clearer understanding:
      If 80 gives us 120, what will 100 give us? Here you have three numbers given in this order: 80 120 100. To find the fourth number which is sought, take lengthwise on the Instrument the second of the given numbers, which is 120, and apply this crosswise to the first, which is 80; then take crosswise the third number, which is 100, and measure that lengthwise along the scale. What you will find-that is, 150-will be the fourth number sought. Notice that the same will happen if instead of taking the second number you take the third, and then in place of the third you take the second; that is, the same will result from the second number taken lengthwise and fitted crosswise to the third and then taking the third crosswise and measuring it lengthwise, as given by the third taken lengthwise and fitted crosswise to the first, thereafter taking the second crosswise and measuring it lengthwise; for in both ways we shall get 150. It is good to notice this, because according to different circumstances one [or the other] way of working will turn out to be the more convenient.
      Some cases may occur in the operation of this rule-of-three that could give rise to some difficulty if we are not aware how to proceed in them. And first, it may sometimes happen that of the three given numbers, neither the second nor the third taken lengthwise can be fitted crosswise to the first, as when we ask "25 gives me 60; what will 75 give?" Here both 60 and 75 exceed double the first number, which is 25, so that neither one can be taken lengthwise and then applied crosswise to 25. Therefore to carry out our intent we shall take either the second or the third lengthwise and fit it crosswise to double the first, that is, to 50; and if double is still not enough, then we shall fit it to triple, or quadruple, etc. Then taking the other one crosswise, we shall affirm that what is shown us by measuring lengthwise the half (or third or fourth part) of what was sought. Thus in the example given, 60 being taken lengthwise and fitted crosswise to the double of 25 (that is, to 50), and 75 being then taken crosswise and measured lengthwise, we shall find that it gives us 90, whose double (or 180) is the fourth number that was sought.
      Besides this, it may happen that the second or third given number cannot be fitted to the first because the first is so large that it exceeds the largest number marked on the Lines, which is 250; as when we say "280 gives me 130; what will 195 give me?" In such cases let 130 be taken lengthwise and put crosswise to the half of 280, which is 140; next take crosswise half the third number (195), or 97 1/2. That distance measured lengthwise will give us 90V2, which is what we sought.
      It will be good to add a further warning, to be used when the second or third of the given numbers is very large, the other two being of medium size, as when we say "If 60 gives me 390, what will 45 give me?" Take 45 lengthwise and fit it crosswise to 60, but then, being unable to take 390 whole, we shall take whatever part of it we like. For example I shall take 100 crosswise, which measured lengthwise will give me 75, and since 390 is 90 taken once and 100 taken three times, 1 shall take the 75 (already found) three times, and add the 67 1/2 that was found for 90; the sum is 292 1/2, the fourth number that was sought.
      Finally we may say also how the same rule operates for very small numbers even though on the Instrument we were able to mark only the points from 15 on, because of the hinge that joins and unites the arms of the Instrument. On this occasion we shall use tenths of the graduations as if they were units when saying (for example) "if 10 gives 7, what will 13 give?" Being unable to take 7 and fit it to 10, we shall take 70 (that is, 7 tens) and fit this to 10 tens (that is, to 100). Next taking 13 tens [crosswise], we measure that distance lengthwise and find it to contain 91 graduations, which means 9 1/10, we having (as said) made every tenth count for one unit.
      From all these instructions, well practiced, you will be able to find easily the solution in every case of any difficulties that may arise.

THE INVERSE RULE-OF-THREE RESOLVED BY means of the same lines.

Operation V.

y operations not dissimilar are resolved questions of the inverse rule-of-three; here is an example. What [amount of] food that suffices for 60 days to maintain 100 soldiers will sustain as many for 75 days? These numbers arranged by the rule will stand in this order: 60 100 75. Operation on the Instrument requires that you take lengthwise the first number, which is 60, and apply it crosswise to the third number, which is 75; then without moving the Instrument you take crosswise 100, which is the second number, and measure this lengthwise, finding 80, which is the number sought. Here you should notice also that we would find the same by fitting the second lengthwise to the third crosswise and then measuring [lengthwise] the first, taken crosswise. You should also note that all instructions given above for the rule-of-three are again to be exactly followed in this.

RULE FOR MONETARY EXCHANGE.

Operation VI.

y means of these same Arithmetic Lines we can change every kind of currency into any other, in a very easy and speedy way. This is done by first setting the Instrument, taking lengthwise the price in the money we want to exchange, and fitting this crosswise to the price in the money into which exchange is to be made. We shall illustrate this by an example so that everything is clearly understood. For instance we want to change [Florentine] gold scudi into Venetian ducats; since the price or value of the ducat is 6 lire 4 soldi, it is necessary (because the ducat is not an exact multiple of the lira and those 4 soldi, enter in) that we work out both currencies in terms of soldi, considering that the scudo is priced at 160 soldi, the price of the ducat being 124. Hence to set the Instrument for changing gold scudi into ducats, take lengthwise the value of the scudo, which is 160, and opening the Instrument fit this crosswise to the value of the ducat, which is 124. Then, not moving the Instrument again, whatever amount is given in scudi and is to be changed into ducats is taken crosswise and measured lengthwise. For example, we want to know how many ducats make 186 scudi; take 186 crosswise and measure this lengthwise. You will find 240, and that many ducats will make 186 scudi.

RULE FOR COMPOUND INTEREST, ALSO CALLED

"New Year's gain."

Operation VII.

e can very speedily solve questions of this kind with the aid of the same Arithmetic Lines, and operating in two different ways, as will be made clear and evident by the two following examples. It is asked what will be gained on 140 scudi in five years at the rate of 6 percent per annum, leaving the interest on the capital and on the previous interest so that all continually go on earning. To find this, take the initial capital (that is, 140) lengthwise and fit this crosswise at 100-100. Then without moving the Instrument take crosswise the distance between points 106106, which is 100 with [a year's] interest, widening the Instrument to fit this distance taken on a compass and applied to 100-100. Next, opening the compass a bit more, take with it crosswise the distance that is now between points 106-106, and again widening the Instrument a little, fit the distance just found to 100-100. Once again open the compass to take this 106-106, and in brief go on repeating the same operation as many times as the number of years of earning; thus in the present example the earnings for five years mean repeating the operation five times. Finally, measuring lengthwise the interval you shall have reached, you will find this to contain 187 1/3 graduations, and that is the number of scudi which the original 140 have become by compounding at 6 percent for five years. Observe that when it would be more convenient to use 200 and 212 in place of 100 and 106, as often happens, the result will be the same.
      The second way of working requires no change in setting of the Instrument after its initial adjustment; it goes as follows, using the same problem as before. To set the Instrument take 100 together with the first [year's] interest, making 106, lengthwise; open the Instrument and fit this crosswise to 100-100, never changing the Instrument again. Then take the amount of money, which was 140, measure this lengthwise, and you will see that the capital and increase after the first year is 148%. To find [this for] the second year, take this 148 2/5, crosswise and (of course) measure it lengthwise; you will find 157 1/3 for the second year. Next take this number 157 1/3 crosswise, again measure lengthwise, and you will find 166 3/4 for capital and earnings at [the end of] the third year. Take this 166 3/4 crosswise and measure it lengthwise, and for the fourth year you will have 176 3/4; finally, take this crosswise and again measure lengthwise, and for the fifth year you will have 187 1/3 as capital and earnings. For more years, if you wish them, go on repeating the operation. Notice that if the original capital is a sum that exceeds 250, [the largest] marked on the Arithmetic Lines, you must operate by parts, taking a half, a third, a quarter, a fifth, or any other part of the given sum; at the end, by taking two, three, four, five, (or more) times what you have found, you will know the desired amount.