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Indeed, this is what Galileo is credited to have done. This study aims to show that the experimental confirmation of his conjectural ideas was a constant concern for Galileo from the very beginning of his scientific activity. In order to support my view, I shall quote his early experimental work and first show the extent to which he was able to analyse his results in depth by himself. Then I shall give evidence that he discussed his results with other scientists to help confirm his initial intuitive ideas. He consulted with them and their advice helped him to come to his final conclusion regarding his insight.

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He also included a description of a hydrostatic balance that determined the precise composition of an alloy of two metals. Some authors have written that he proceeded by immersing the crown in water, having previously and separately immersed equal amounts [in weight] of very pure gold and silver, and, from the differences in their making the water rise or spill over, he came to recognize the mixture of gold and silver of which the crown was made.

I may well believe that, a rumor having spread that Archimedes had discovered the said theft by means of water, some author of that time may have then left a written record of this fact; and that the same [author], in order to add something to the little that he had heard, may have said that Archimedes used the water in that way which was universally believed.

But my knowing that this way was altogether false and lacking that precision which is needed in mathematical questions made me think several times how, by means of water, one could exactly determine the mixture of two metals. And at last, after having carefully gone over all that Archimedes demonstrates in his books On Floating Bodies and Equilibrium, a method came to my mind which very accurately solves our problem. I think it probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself.

This method consists in using a balance whose construction and use we shall presently explain, after having expounded what is needed to understand it. One must first know that solid bodies that sink in water weigh in water so much less than in air as is the weight in air of a volume of water equal to that of the body. This [principle] was demonstrated by Archimedes, but because his demonstration is very laborious I shall leave it aside, so as not to take too much time, and I shall demonstrate it by other means.

Let us suppose, for instance, that a gold ball is immersed in water. If the ball were made of water it would have no weight at all because water inside water neither rises nor sinks. It is then clear that in water our gold ball weighs the amount by which the weight of the gold [in air] is greater than in water. The same can be said of other metals. And because metals are of different [specific] gravity, their weight in water will decrease in different proportions. Let us assume, for instance, that gold weighs twenty times as much as water; it is evident from what we said that gold will weight less in water than in air by a twentieth of its total weight [in air].

Let us now suppose that silver, which is less heavy than gold, weighs twelve times as much as water; if silver is weighed in water its weight will decrease by a twelfth. Thus the weight of gold in water decreases less than that of silver, since the first decreases by a twentieth, the second by a twelfth. Let us suspend a [piece of] metal on [one arm of] a scale of great precision, and on the other arm a counterpoise weighing as much as the piece of metal in air.

If we now immerse the metal in water and leave the counterpoise in air, we must bring the said counterpoise closer to the point of suspension [of the balance beam] in order to balance the metal.

Let, for instance, ab be the balance [beam] and c its point of suspension; let a piece of some metal be suspended at b and counterbalanced by the weight d. If we immerse the weight b in water the weight d at a in the air will weigh more [than b in water], and to make it weigh the same we should bring it closer to the point of suspension c, for instance to e. As many times as the distance ac will be greater than the distance ae, that many times will the metal weigh more than water.

Come, per essempio, sia la bilancia ab, il cui perpendicolo c; ed una massa di qualche metallo sia appesa in b, contrapesata dal peso d. Let us then assume that weight b is gold and that when this is weighed in water, the counterpoise goes back to e; then we do the same with very pure silver and when we weigh it in water its counterpoise goes in f. This point will be closer to c [than is e], as the experiment shows us, because silver is lighter than gold.

The difference between the distance af and the distance ae will be the same as the difference between the [specific] gravity of gold and that of silver. But if we shall have a mixture of gold and silver it is clear that because this mixture is in part silver it will weigh less than pure gold, and because it is in part gold it will weigh more than pure silver.

If therefore we weigh it in air first, and if then we want the same counterpoise to balance it when immersed in water, we shall have to shift said counterpoise closer to the point of suspension c than the point e, which is the mark for gold, and farther than f, which is the mark of pure silver, and therefore will fall between the marks e and f.

From the proposition in which the distance ef will be divided we shall accurately obtain the proportion of the two metals composing the mixture. So, for instance, let us assume that the mixture of gold and silver is at b, balanced in air by d, and that his counterweight goes to g when the mixture is immersed in water.

I now say that the gold and silver that compose the mixture are in the same proportion as the distances fg and ge.

We must however note that the distance gf, ending in the mark for silver, will show the amount of gold, and the distance ge ending in the mark for gold will indicate the quantity of silver; so that, if fg will be twice ge, the said mixture will be of two [parts] of gold and one of silver.

And thus, proceeding in this same order in the analysis of other mixtures, we shall accurately determine the quantities of the [component] simple metals. Suspend it in its middle point; then adjust the arms so that they are in equilibrium, by thinning out whichever happens to be heavier; and on one of the arms mark the points where the counterpoises of the pure metals go when these are weighed in water, being careful to weigh the purest metals that can be found.

Having done this, we must still find a way by which easily to obtain the proportions in which the distances between the marks for the pure metals are divided by the marks for the mixtures. This, in my opinion, may be achieved in the following way. Thus, for instance, on the marks e, f I wind only two turns of steel wire and I do this to distinguish them from brass ; and then I go on filling up the entire space between e and f by winding on it a very find brass wire, which will divide the space ef into many small equal part.

When then I shall want to know the proportion between fg and ge I shall count the number of turns in fg and the number of turns in ge, and I shall find, for instance, that the turns in fg are 40 and the turns in ge 21, I shall say that in the mixture there are 40 parts of gold and 21 of silver. Sopra i termini de i metalli semplici avvolgasi un sol filo di corda di acciaio sottilissima; ed intorno agli intervalli, che tra i termini rimangono, avvolgasi un filo di ottone pur sottilissimo; e verranno tali distanze divise in molte particelle uguali.

Here we must warn that a difficulty in counting arises: Since the wires are very fine, as is needed for precision, it is not possible to count them visually, because the eye is dazzled by such small spaces. To count them easily, therefore, take a most sharp stiletto and pass it slowly over the said wires. Thus, partly through our hearing, and partly through our hand feeling an obstacle at each turn of wire, we shall easily count said turns.

And from their number, as I said before, we shall obtain the precise quantity of pure metals of which the mixture is composed. Note, however, that these metals are in inverse proportion to the distances: Thus, for instance, in a mixture of gold and silver the coils toward the mark for silver will give the quantity of gold, and the coils toward the mark for gold will indicate the quantity of silver; and the same is valid for other mixtures.

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