ON THE OPERATIONS OF THE QUADRANT.

y attaching the Quadrant Arc to the Instrument we have the gunners' Squadra, divided as is customary into 12 "points." The ordinary use of this is to place one of its legs in the bore of a cannon, having first hung a plumb bob by a thread from the pivot of the Instrument; this thread cutting the circumference, we are shown the elevation of the cannon in "points" as 1, 2, or 3. But since to use the Squadra in this way is not without peril, [the gunner] having to go outside the ramparts or shields and expose himself to the enemy, another way has been thought of to do the same thing in safety-that is, by applying the Squadra near the fuse of the cannon [to its upper surface]. But because the internal bore is not parallel with the outer surface, the metal being thicker near the breech, one must provide against error by lengthening the leg- of the Squadra on the side toward the mouth, adding to this [length] by a movable footing. Thus, placing the Squadra [upright] near the fuse [of a level cannon], we lengthen the forward leg by this footing until the vertical [thread] cuts point 6 [on the usual scale of 12]; and fixing the footing with its set-screw, we mark a little line on the side of the Instrument at the end of the cursor carrying the footing, so that on every occasion of its use [with this cannon] we can place it exactly. Then when we want to fire "at one point" of elevation, the cannon is elevated until the thread cuts the number 7; and if we want "two points" it must cut the 8, and so on.
      Next following is the division of the Astronomical Quadrant, whose use (having been treated by many. others) will not be explained here. The circumference that comes next is seen to be divided by some slanted lines, used for taking the slope of any [rampart] wall, beginning with those which for every ten units of height have one unit of horizontal advance and going on out to those which have one unit of horizontal advance for every 1 1/2 units of height. To employ this [scale of the] Instrument we must hang the [plumb] line from that little hole which is seen at the beginning of the gunners' Squadra;" then, approaching the sloping wall, hold the other edge of the Instrument against it and note where the thread cuts [this scale] ; if it cuts at number 5, for example, we shall say that the wall has one bracclo of horizontal advance for every 5 braccia of height; or, cutting at number 4, that it has one unit of horizontal advance for every 4 of height."

VARIOUS WAYS OF MEASUREMENT BY SIGHTING; and first, of Vertical Heights whose bases we can approach, and [from which we can] retire.

he final circumference, divided into 200 parts, is a scale for measuring heights, distances, and depths by means of sighting. And beginning first with heights, we shall show various ways of measuring them, starting with vertical heights whose bases we can reach, as it would be if we wished to measure the height of the tower AB. Being at point B, let us move toward C and walk 100 paces or other units. Stopping at C, let us sight along one side of the Instrumentat the height A, as is seen for side CDA, and note the graduations cut by the thread DI. If these fall in the 100 away from the eye, as in the given example for arc 1,22 the number of those graduations is the number of paces (or other units we have measured along the ground) that we shall say are contained in height AB. But if the thread cuts the other 100, as seen in the next diagram in which we want to measure the height GH, our eye being at I and the thread cutting points M and 0, then take that number of graduations and divide it into 10,000; the result will be the number of units contained in height GH. For example if the thread cuts point 50 [in the arc near the eye], then divide 50 into 10,000 and get 200, and that many units [of which 100 equal IH] will be contained in height GH.
      We have seen that sometimes the thread will cut the 100 away from the side along which we sight, and sometimes it will cut the 100 touching that side; either may happen in many of the ensuing operations. Therefore as a general rule it is always to be remembered that when the thread cuts the first 100, contiguous to the sighting side, one must divide 10,000 by the number cut by the thread, following in the rest of any operation whatever rule is written there; for in the ensuing examples we shall always assume that the thread cuts the second 100 [away from the eye].
      The better to make clear the multitude of uses of this Instrument of ours, I wish to simplify the more laborious calculations required by the rules for measurement by sighting; they can be performed without trouble and very swiftly by using a compass along the Arithmetic Lines. But starting from the above operation, for those who cannot divide 10,000 by a number cut by the vertical [thread], I say that you always take 100 lengthwise along the Arithmetic Lines and fit this crosswise to the number of graduations cut by the vertical; then without altering the Instrument you take the distance between points 100-100, which next measured lengthwise will give you the altitude sought. For instance if the thread has cut at 77, take 100 lengthwise along the Arithmetic Lines, fit this crosswise to points 77-77, at once take crosswise the distance between points 100-100, and measure that lengthwise. You will find it to contain 130 graduations, and that many units [of those paced off ] you will say to be contained in the height we wished to measure.
      We shall be able to measure such a height in another way without the necessity of stepping off 100 units of ground, as will be clearly shown. If, for example, we should wish to measure the height of the tower AB from point C, we could point the side CDE of the Instrument at the summit A, noting the graduation cut by the thread EI, and let this be, say, 80. Then, without changing place, we merely lower the Instrument and sight at some low mark placed on the tower, such as point F, noting the number of the graduation now cut by the thread, say 5. Next see how many times this smaller number, 5, goes into the other, 80, which is 16 times, and we shall say that distance FB is contained 16 times in the whole height BA. And since point F is low, we can easily measure FB with a staff or the like, and thus come to know the altitude BA. Notice that in measuring heights we find and measure only the height above our visual horizon[tal], so that when the eye is above the root or base of the thing being measured one must add, to the height found by using the Instrument, as much more as the eye is above the base.
     The third way to measure such a height will be by rising or descending. Thus, wishing to measure height AB, hold the Instrument at some place above the ground, as at point F, and sight at point A along side EF, noting the graduations between G and I as cut by the thread to be, say, 65. Then, going down and being vertically under point F, say at point C, we sight the same height along side DC, noting the graduations between L and 0, which will be more than before, as say 70. Then take the difference between those numbers 65 and 70, which is 5, and as many times as this is contained in the larger number (that is, in 70, which will contain 5 fourteen times), that many times will height BA contain distance CF, which latter we shall measure, as we can easily do, and thus we shall come to know the height of AB.
     If we want to measure a height whose base cannot be seen, such as the altitude of the mountain AB, we being at point C, let us sight at the summit A and note the graduation at I cut by the vertical DI, and for example let this be 20; next, approaching the mountain 100 paces and coming to point E, we sight the same summit and note graduation F, which is 22. This done, multiply together the numbers 20 and 22, making 440, and divide this by the difference between those same numbers, which is 2, giving 220, and that many paces high we shall say the mountain is. 
     The calculation can be done on the Instrument by taking the smaller number of graduations lengthwise along the Arithmetic Lines and fitting this crosswise at [the numbers representing] the difference between the two numbers of graduations; then take crosswise the larger number of graduations, which measured lengthwise will give us the altitude sought. If, for instance, the graduations cut were 42 and 58, take 42 lengthwise and fit it crosswise to the difference of the said numbers-that is, at 16-16. If unable to do this, use the double, triple, or quadruple; say [you use] the quadruple, which is 64. Then take 58, or [rather] its quadruple 232, and measure this lengthwise, which will give you 152 1/4, the height sought.
      Moreover we can measure with the same Instrument one height placed on top of another, as when we would measure the height of tower AB on top of mountain BC. First, from point D let us sight the top of the tower, A, and note the graduation cut by the thread EI, which for example shall be 18. Then, leaving a staff standing at point D, we advance until the base of the tower, B, is sighted with the vertical GO again cutting the previous number, 18, say when we have got to point F. Then measure the paces between these two stations, D and F, which for example let be 130; multiply that number by the previous 18, making 2,340, which you divide by 100, getting 23%, the height of the tower in paces.
      On the Instrument the computation will be done by taking lengthwise the number of paces (or that of graduations) and fitting it at points 100-100; then take the other number crosswise and measure this lengthwise. For example if the graduation cut was 64 and the paces were 146, take 64 lengthwise and fit it crosswise to 100-100; then take 146 crosswise and measure this lengthwise, giving us approximately 93 1/2, which is the height sought. As to depths, we shall have two ways of measuring these. The first will be to measure a depth contained between two parallel lines, like the depth of a well or the [inside] height of a tower from its top. Thus, let there be the well ABDC between the parallel lines AC and DB. With the corner of the Instrument toward the eye E, sight alongside EF in such a way that the visual ray passes through points B and C, noting the number cut by the thread, which for example let be 5. Then see how many times this number goes into 100, and that many times will the breadth BA [which can me measured] be contained in the depth BD.
      The other way will be to measure a depth of which the [vertical] base cannot be seen, as when we are on the mountain BA and want to measure its height above the surrounding plain. In such a case we raise ourselves above the mountain, climbing up some house, tower, or tree; and placing the eye at point F we sight at some mark situated in the plain, as at point C, noting the graduation cut by the thread FG, say 32. Then, descending to point D, we sight the same mark C along side DE, noting the graduations along AI, which let be 30. Take the difference between these two numbers, which is 2, and see how many times this goes into the smaller number, which is 15 times; then we say that the height of the mountain is 15 times the height FD, which we can measure, and this will give us what we sought.

     We pass on now to the measurement of distances, such as the width of a river from one of its banks or some other high place, as seen in the diagram. We want to measure the width CB. From point A we sight the [river] edge B along side AF, noting the graduation in DE cut by the vertical, which shall for example be 5. However many times this number goes into 100, that many times will the height AC go into the width CB; so measuring the height AC and multiplying it by 20 we shall have the width sought.
     We can measure such distances in another way. Being at point A, for example, we want to find the distance to point B. Set the Instrument at right angles, one leg being pointed toward B, and sight along the other leg toward point C. Measure 100 paces (or other units) in the direction AC and leave a staff driven at point A, placing another at C. Then, at C, point one leg of the Instrument toward A and sight through corner C at mark B, noting on the Quadrant the point cut by the visual ray, which shall be point E. Divide that number into 10,000; what you get will be the number of paces (or other units) between point A and mark B.
      But if we are not permitted to move the 100 paces along a line at right angles to the first sighting, we must proceed otherwise. For example we are at point A and want to take the distance D, but cannot move except along AE which makes an acute angle with AB. In order to carry out our intent we shall first aim one leg of our Instrument along this path, as seen by line AF, and without changing that setting we sight point B through angle A, noting the graduation cut by ray AD, which for example is 60. Then, leaving a staff at point A, we shall put another one 100 paces along line AE, at point F. There we hold the corner of the Instrument, aiming leg EF at staff A; we sight mark B from corner F, noting the graduations [between] G and I to be, for example, 48. To find distance AB from these numbers 60 and 48, multiply the first by itself, making 3,600; add 10,000 and get 13,600; take the square root of that number, which will be about 117; multiply this by 100 to make 11,700; finally, divide that number by the difference between the two original numbers 60 and 48 (that is, by 12). The result is 975, which is undoubtedly the distance AB, in paces.
      Calculation for this operation is performed on the Instrument as explained in the following example. Let the graduations cut by the two visual rays be 74 and 36. To make the computation, first set the Instrument so that the Arithmetic Lines are at right angles, which may be done by taking 100 graduations lengthwise on these and fitting this by a compass across the same Lines in such a way that with one of the [compass] legs at point 80, the other falls at 60 (and this rule for setting the lines at right angles should be kept in mind for other uses). Then take the crosswise distance between point 100 and the larger of the two numbers cut by the rays, which was 74; this distance must then be fitted crosswise to the difference between the two numbers of graduations cut by the rays, which is 38. If this cannot be done because of the smallness of the number, then use its double, triple, or quadruple; here, for example, apply its triple, which is 114, and then take the crosswise distance between points 100-100. Measure that distance lengthwise; taken three times, this will be the distance sought. Measured in the present example, you will find it to be 109, so that tripled it will give you 327, which is approximately the distance we wanted to measure.
      Next let us see how to measure the distance between two places remote from us, and first we shall explain the procedure when from some place we can see them both along the same straight line. This is done in the present example, where we wish to measure the distance between points A and B, which from point C appear along the same line CBA. First set one leg of the Instrument in that direction', and sight [perpendicularly] along the other leg toward D, where a staff is placed 100 units from C, and a similar one at C. Going to point D, we hold one leg of the Instrument in the direction DC, sighting from corner D the two places B and A, and noting the numbers cut by these rays, which are, say, 20 and 25. Divide 10,000 by those numbers, and the difference between the two results will be the distance BA.
      But if we want to measure the distance between two places C and D and are unable to reach any place from which they appear in the same direction, we proceed in such cases as next described. Assume that standing at place A, we want to investigate the distance between the two places C and D. First, set one leg of the Instrument as shown by line AEC, sight through an angle the other point, D, and note the graduations [between] E and F (cut by ray AFD), these being for example 20. Without moving the Instrument, sight along the other leg toward B, planting a staff at A and having another one placed along the direction AB. Then, walking in that direction, we get to B, having departed from the second staff just enough so that again pointing one leg of the Instrument along line BA, the other leg points at D (as shown by line BD). From corner B we sight point C, noting the number cut by ray BG, which (say) is 15. Finally, the number of paces between the two stations A and B is measured, which in our example is 160. Now, coming to the arithmetical work, first multiply the number of paces between the two stations (that is, 160) by 100, making 16,000; divide this separately by the numbers of [Quadrant] points-that is, by 20 and by 15-giving us the two numbers 800 and 1,067. Take the difference between these, which is 267, and multiply it by itself, making 71,289. Add that number to the square of the number of paces (160), which is 25,600, for a total of 96,889. Take the square root of this number, which is 311, and that many paces we shall say to lie between the two places C and D.
      How one may find the computation on the Instrument will be clear from the following example. Let the two numbers cut by the rays be, for example, 60 and 34, and the number of paces be 116. For the operation, always take 100 lengthwise along the Arithmetic Lines, apply this crosswise at the greater of the numbers cut by the rays (which here is 60), and at once take crosswise the number of paces (here 116). Fit this distance crosswise to the other number of rays, which here is 34; if you cannot, then fit its double, triple, quadruple, or whatever turns out most convenient. Here let us take the quadruple, which is 136. This done, take crosswise the numerical difference between the two numbers cut by the rays, in this case 26, or take its double, triple, or quadruple according to the application made just previously, whence in this case you should take the quadruple, or 104. Measure this distance lengthwise, keeping in mind the number obtained, which in the present example will be 148. Finally, set the Arithmetic Lines at right angles in the way explained above; that done, take crosswise the difference between the number you kept in memory and the number of paces (that is, between 148 and 116;) this lengthwise and you will find 188, which is exactly the distance DC that was sought.
      Finally, if we cannot move in the way required in the previous operation, we shall nevertheless be able to find the distance between two remote places in this other way. Being at point C, for example, we wish to find the distance between the two places A and B. By using any of the procedures already explained, we shall measure separately the distance between C and A and the distance between C and B; let the former be for example 850 paces, and the latter 530. At mark C, set one leg of the Instrument to point at A (as shown by line CDA), and from corner C sight at the other terminus B, noting the number of graduations [between] D and E cut by this ray, which for example is 15. Multiply that number by itself, making 225, and 10,000 making 10,225; take the square root, which is 101. Multiply the smaller distance, 530, by 100, making 53,000, which you divide by the root just found, getting 525. This you multiply by the greater distance, 850, making 446,250, which number must be doubled, making 892,500. Next you must multiply separately each of the two distances by itself, getting 722,500 and 280,900; add these together, making 1,003,400. Subtract from that the double found before, 892,500, and there remains 110,900, whose square root, which is 347, will be the distance [in paces] between the two places A and B, which was desired.
      With a notable reduction of labor we can make the computation on the Arithmetic Lines in the way made clear by an example. Suppose the longer distance to be 230 paces, the shorter 104, and the number cut by the ray to be 58. Set the Arithmetic Lines at right angles, and placing one leg of a compass at point 100, move the other crosswise to the number of the point cut by the ray, which was 58; then see how much this distance is, measured lengthwise. You will find it to be about 116, to be kept in memory. Then take lengthwise the number 58 (which was the point cut by the ray) and open the Instrument until that distance fits crosswise between 100 [on one arm] and the 116 that you kept in memory [found along the other arm]. Not changing the Instrument, take with a compass the crosswise distance between the two numbers of paces, 230 and 104, and this measured lengthwise finally gives you 150 paces," truly the distance AB.
      Gentle Reader, I have judged it sufficient for now to have described these rules for measurement by sighting-not that they alone can be worked on the Instrument, there being a great many others-but in order not to engage unnecessarily in long discourses, I being certain that anyone of average intelligence will have understood those already explained and can find others for himself, suited to any particular case that may arise.
      Not only might I have gone on much longer about rules for measurement by sighting, but I could expand on many, many more rules by showing the solution of (I may say) infinitely many other problems of geometry and arithmetic which can be solved with the other Lines of our Instrument; for as many problems as there are in Euclid's Elements and in other authors, they are thus resolved by me very quickly and easily. But as was said at the outset, my present intention has been only to speak to military men, and of few things beyond those concerning that profession, reserving to another occasion the publication of the construction of the Instrument and a more ample description of its uses.

THE END.