EXPLANATION OF THE METALLIC LINES, MARKED NEXT TO THE STEREOMETRIC.

Operation XXI.

he present Lines have divisions to which are affixed these symbols: Au, Pb, Ag, Cu, Fe, Sn, Mar, Sto, which mean Gold, Lead Silver, Copper, Iron, Tin, Marble, and Stone. From these you can get the ratios and differences of [specific] weight found between the materials thus designated, in such a way that with the Instrument set at any opening, the intervals between any correspondingly marked pair of points will give the diameters of balls (or sides of other solid bodies) similar to one another and equal in weight. Thus, whatever be the weight of a ball of gold whose diameter is equal to the distance Au-Au, such is also the weight of a lead ball whose diameter is the distance between points Pb-Pb, and of a marble ball whose diameter is the distance between points Mar-Mar [at the same setting].

     From this we may know at once how large to make a body of any one of the above materials so that it will weigh equally with another, similar in shape but made of another of the said materials-an operation that we shall call 96 transmutation of material." For instance, if line A is the diameter of a tin ball and we want to find the diameter of a gold ball equalling it in weight, we take with the compass line A, and opening the Instrument we fit that to the points Sn-Sn; then we immediately take the distance between points Au-Au, and that will be the diameter of the gold ball (represented by line B) which equals in weight the other, of tin. And the same is to be understood of all other solid bodies of the other designated materials. Now if we combine the use of these Lines with that of the previous Lines, we shall collect therefrom many important advantages, as will be explained below. And first:

HOW WITH THE AFORESAID LINES WE CAN find the ratio of weight between all metals and other materials marked along the Metallic Lines.

Operation XXII.

e wish, for example, to find the ratio of [specific] weight between the two metals silver and gold. With a compass take the distance between the pivot of the Instrument and the point marked Ag; with the Instrument opened, fit this [crosswise] to any number you please along the Stereometric Lines, say for example to points 100-100. Then, without altering the Instrument, take the distance between the pivot and the point Au and see what number this fits along the Stereometric Lines; finding, for example, that it fits at points 60-60, you will say that the ratio of specific weight of gold to silver is as 100 to 60. Notice that in operation, the diameters taken and fitted to the Stereometric Lines exhibit inversely the ratio of weight for the metals, so that as seen from the above example the diameter of the silver gives you the weight of the gold, while the diameter of the gold gives the weight of the silver. Thus we come to understand how gold is heavier than silver in the ratio of 40 percent, since 40 is the difference between the two weights found, for gold and silver [gold being 100].
      From this we learn the solution of a very pretty question, which is: Given any shape of one of the materials marked along the Metallic Lines, to find how much of some other of the said materials will be needed for forming another solid equal to it. For instance, we have a marble statue and we want to know how much silver would go to make one of the same size. To find this weigh the marble statue, and suppose its weight to be 25 pounds; then take the distance between the pivot of the Instrument and point Ag, the material of the future statue, and opening the Instrument fit that distance [crosswise] to the Stereometric Lines at the points marked with the number of the weight of the statue (that is, to points 25-25); without changing the Instrument you then take the distance between the pivot and point Mar, and see to what number of the Stereometric Lines it fits crosswise. Finding that it fits at points 96-96, you will say that 96 pounds of silver are required to make a statue equal in size to that of the marble one.

COMBINING THE USES OF THE METALLIC AND the Stereometric Lines, and given two sides for two similar solids made of different materials, to find the ratio of their weights.

Operation XXIII.

ine A is the diameter of a copper ball and B is the diameter of one of iron; we want to know the ratio of their weights. Take line A with a compass, and opening the Instrument fit this to the points of the Metallic Lines marked Cu-Cu. Without changing this setting, immediately take the distance between points Fe-Fe, which shall be line X. Now, X being unequal to B, and being the diameter of an iron ball equal in weight to A, it is evident that the difference [in diameter] between the two balls A and B will be the same as the difference between X and B. Since X and B are of the same material, their difference is easily found with the Stereometric Lines, as explained before in Operation 16; that is, we shall take line X and (opening the Instrument) fit it to some number, as for instance 30-30, which done we shall see at what point line B fits. Finding it, for example, to fit at 10-10, we shall say that the copper ball [of diameter] A is triple that of the iron ball [of diameter] B.

     The converse of the foregoing operation can be done equally easily with the same Lines, that is: Given the weight and the diameter (or side) of a ball (or other solid) of one of the materials marked on the Instrument, how one may find the size of another similar solid of some other of the said materials which will weigh equally with any given weight. For example, line X being the diameter [this time] of a marble ball weighing 7 pounds, find the diameter of a lead ball that weighs 20 pounds. Here it is seen that we must perform two operations; one is to transmute marble into lead, while the other is to increase the weight of 7 to 20. The first operation is performed with the Metallic Lines, fitting diameter X crosswise to points Mar-Mar, and then, without changing the Instrument, taking the distance between points Pb-Pb, which will give the size of the lead solid that weighs as much as the given marble, or 7 pounds. But since we want 20 pounds, we have recourse to the aid of the Stereometric Lines; and fitting this distance to points 7-7 thereon, we at once take the crosswise distance between points 20-20, which will equal line D; and doubtless that will be the side of the solid lead figure that will weigh 20 pounds.

HOW THESE LINES SERVE UNIVERSALLY FOR the Gunners' calibration for all cannonballs of any material and of every weight.

Operation XXIV.

t is a very manifest thing that the weight of different materials is different and that iron is I specifically] much heavier than stone, and lead than iron; from which it follows that artillery pieces being fired now with stone balls, again with iron, and other times with lead balls, the same cannon that carries so much [weight] of lead ball carries less [weight] of iron, and still less of stone, whence different charges must be used for the different balls. Therefore those tables for firing (or "calibers") in which are shown the diameters and weights of iron balls cannot be used also for stone balls, but [for the same weight] the diameters must be increased or diminished according to different materials. Moreover it is manifest that the [standard] weights used are different in different countries, and in not only every province, but in almost every cityfrom which it follows that the calibration suited to the weights of one place cannot be used for the weights of another, and according as the "pound" shall be greater or less in one place than another, it will be necessary for fixed divisions of caliber to carry [shots] longer or shorter distances. From which we may conclude that a caliber adapted to any kind of material and every difference in weight must necessarily be variable and capable of increase or decrease. Such is precisely what is marked on our instrument; for opening this more, or less, the intervals between the divisions found on it increase or diminish without in any way changing their ratios.
      Having stated these things generally, let us pass to the particular application of this calibration to any differences in weight and to all the various materials. And since one can come to knowledge of anything unknown only by means of something else, known, it is necessary to know [in our case] only the diameter of a single ball, of any material whatever, and any weight that corresponds to the "pounds" customary in the country where we wish to use our Instrument. From that single diameter we shall, by means of our calibration, come to know the weight of any other ball, of whatever other material-meaning, however, materials marked on our Instrument.
      Now the manner of obtaining such knowledge may be made manifest easily by an example. Suppose for instance that we are at Venice, and we wish to make use of our calibration to know the range of some artillery pieces. First we get the diameter and the weight of a ball made of any of the materials designated on our Instrument; suppose for instance that we have the diameter of a 10-pound lead ball in Venetian weight. This diameter we shall mark by two points [scratched] along the side of one arm of the Instrument. Then when we wish to accommodate and adjust our calibration in such a way that by taking the bore of a [Venetian] piece of artillery and carrying this [measurement] to our calibration, we learn how many pounds of lead ball this [gun] carries, we need only take on a compass that diameter of ten pounds of lead (already marked along the side of our Instrument) and then open the Instrument until the said diameter fits to points 10-10 of the Stereometric Lines. Set in that way, those Lines will serve us for exact calibration, so that given the diameter of the mouth of any artillery piece, and transferring it to the said calibration, we shall know (from the number of the points to which it fits) how many pounds of lead ball the said cannon will bear. And if we want to set the Instrument so that the calibration shall correspond to iron balls, then we still take the same diameter of ten pounds of lead marked on the side, and fit this [crosswise] to the points marked Pb-Pb on the Metallic Lines; then without changing the Instrument we shall take with a compass the distance between the points marked Fe-Fe, which will be the diameter of an iron ball weighing ten pounds. Opening the Instrument, this diameter will be fitted to the points marked 10-10 on the Stereometric Lines, and then said Lines will be exactly set for calibration of iron balls. By similar operations the Instrument can be set for stone balls.
      Notice that since it is necessary to mark along the side of the Instrument various diameters of balls corresponding to the [standard] weights of various countries, we should in order to avoid confusion always mark diameters of balls of lead weighing ten pounds, which we shall find to be larger or smaller according to the differences in [local] weights. It will not be difficult to mark down these diameters without the necessity of actually finding ten-pound lead balls, by what was taught before in Operation 23. There, given the diameter of a ball of whatsoever weight and made of any material, we saw how one finds the diameter of any other, of different weight and of any material-meaning always those materials marked on the Metallic Lines. Thus, finding ourselves in any country, provided only that we can find a ball of marble or stone or any other material designated on our Instrument, we can instantly get the diameter of a tenpound lead ball [in that country's weight].

HOW, GIVEN A BODY OF ANY MATERIAL, WE can find all the particular measures of one of different material that shall weigh a given amount.

Operation XXV.

mong the uses that can be drawn from these same lines, one is that we may increase or diminish solid shapes according to any desired ratio, changing the material or leaving it unchanged, as will be understood from the ensuing example. We are given a little tin model of a cannon from which we must derive all the particular measurements for a large copper cannon weighing, say, 5,000 pounds. First we shall weigh our little tin model; let its weight be 17 pounds. Next we take one of its measurements, whichever you prefer; let this be for example the thickness of the cannon at its mouth. Opening the Instrument, we apply this to the points Sn-Sn of the Metallic Lines, tin being the material of the given model; and since the large cannon is to be made of copper, we at once take the distance between points Cu-Cu, which will be the thickness at the mouth of a copper cannon that would weigh the same as the tin one. But since it is to weigh 5,000 pounds, and not 17 like this tin one, we next have recourse to the Stereometric Lines and fit that distance (just found between the points Cu-Cu) to the points marked 17-17. Without moving the Instrument we then take the distance between points 100-100, which will give the thickness at the mouth of a [copper] cannon weighing 100 pounds. But we want the weight to be 5,000 pounds, whence this distance must be increased in the ratio of 50:1. Therefore, opening the Instrument farther, we put this [last distance] at some number for which there is another [on the Stereometric Lines] fifty times as great, as would be the case if we fitted it to the points 2-2 and then took the distance between points 100-100, which would doubtless be the measure of the thickness that must be given at the mouth. In this order will be found all the particular measurements for all other parts, such as the throat, the trunnions, the breech, and so on.
      No less can we find the length of the cannon, although we cannot open our Instrument to any such distance. To find this, we take on our little model not the entire length, but only a part, such as one-eighth or one-tenth or the like; this may be increased in the way already explained, to show us ultimately one-eighth or one-tenth of the whole length of the large cannon.
      But in this [operation] another difficulty [about metals] may arise. Yet our Metallic Lines, designed so as to give measures for transmuting one simple metal into another, can do the same for an alloy of metals when in the above example we must make the cannon not of pure copper, but of metal mixing copper and tin, as it is the common custom to do. Wherefore, for complete satisfaction, we shall show how, with the aid of those same Metallic Lines, we can find the measurements for any alloy just as for a simple metal. This will be done by adding two tiny points along the Metallic Lines-and I mean very tiny, so that we may remove them after they have served our purpose. Suppose, for example, that the cannon we want to make is to be not of copper (as previously assumed), but must be cast from bronze, in an alloy of three parts copper to one of tin. Then we must carefully divide, on both arms, that short line between the points marked Cu and Sn into four parts, leaving three of these on the side Sn and only one toward Cu, and precisely there making our tiny point appear. We then use this point (marked, as was said, on both the Metallic Lines) for our transmutation of metal, just as above we used the points Cu-Cu. Using this rule we can, as occasion arises, mark new points for any alloys of two metals in any ratio desired.
      It will not be irrelevant and without useful purpose to note, especially when one must make the transmutation into some metal mixed and alloyed of two others in any ratio, that when a single one of the measurements sought has been found by working with great precision in the way explained above, one may on the strength of this unique determined measurement go on to find all the rest by means of the Arithmetic Lines [alone], in a way not much different from that explained in Operation 3. For example, line A was the diameter or thickness of the mouth of the given cannon model, and line B was then found

to be the mouth of the 5,000-pound cannon to be made of a metal containing three parts of copper to two of tin. I say next that to find all the remaining dimensions, we may use the Arithmetic Lines, taking line B and applying it crosswise to any point along those Arithmetic Lines, and the greater the number chosen the better it will be. So let us fit B to the last point, that is, at 250-250, and without moving the Instrument let us see where line A fits crosswise; say this is at 44-44. From this we learn that since A of the model measures 44 graduations, what must correspond to it in the real cannon will be 250 of those same graduations. This same ratio must then hold for every other measurement, whence to find (for example) the thickness of the throat of the real cannon you will take that thickness from the little model and fit it crosswise to points 44-44 of the Arithmetic Lines, taking then (also crosswise) the distance between points 250-250, which will give the throat of the large cannon. By the same rule will be found all the other measurements.
      Moreover, to find first with the utmost precision line B, corresponding to the point for the alloy of the two metals assigned," one may proceed thus. First represent separately the two simple metal measures corresponding to tin and copper as the two lines CD and CE, of which CD is the measure corresponding to pure copper and CE to pure tin. Their difference, being line DE, is to be

divided according to the assigned ratio of the alloy, whence for three parts copper to two parts tin you will cut line DE at point F so that FE (towards tin) is two parts, and FD (towards copper) is two parts. This is done by dividing all line DE into five parts, leaving three of these towards E and two towards D [by marking F]. Line CF will now be our chief one, as line B was before, in proportion to which, by simply using the Arithmetic Lines in the manner taught in Operation 3, all the other measurements will be found without recourse again to the Metallic and Stereographic Lines.