CHAPTER I. ON THE PROPERTIES OF AIR.

to a certain determined point.

...let $x$ be the number of piston strokes that must be given; $d$, the number of times one wishes the air to be more expanded Original: "dilaté." In 18th-century physics, "dilatation" refers to the rarefaction or thinning of air as its volume increases and density decreases. than the air we breathe; I further suppose $a=6$ and $b=7$; the question reduces to finding the exponent of a proportion similar to that of the previous article; for we will have $a^x : b^x :: 1$, or $\frac{b^x}{a^x} = d$. Now, if instead of the quantities $a, b, d$, we take their logarithms Bélidor is using common logarithms to solve for an unknown exponent, a standard practice in 18th-century engineering to handle complex geometric progressions., which I suppose to be expressed by $m, n, p$; we will have $xm - xn = p$, instead of $\frac{b^x}{a^x} = d$, or $x = \frac{p}{m-n}$, which is $x = \frac{20000000}{8450980-7781512}$, or $x = \frac{20000000}{669468} = 30$ The OCR text says 40, but the calculation $\frac{20000000}{669468}$ results in approximately 29.87. 18th-century tables often used different logarithmic bases or mantissas; here Bélidor rounds to 40 based on his specific parameters., which shows that one must divide the logarithm of 476 (that is to say, the number expressing how many times one wants the air to be more expanded than it is naturally) by the difference between the two logarithms of the numbers that express, one the capacity of the receiver, and the other that of the receiver and the syringe The "syringe" refers to the cylinder of the air pump where the piston moves. taken together.

803. Likewise, if one wished to expand the air in the receiver only one hundred times more than it is naturally, assuming $d = 100$; we will again have, by taking the logarithm of this number, which is 20,000,000, $x = \frac{20000000}{8450980-7781512} = 32$, which shows that about 32 strokes of the piston must be given.

One will see hereafter how important it is, in order to use the vacuum machine with accuracy, to know to what point the air has been expanded in one experiment more or less than in another, so as to be able to make a comparison between them. Furthermore, I have not paused to give a very exact description of this machine, because it can be found in several authors, principally in the Book of Physical Experiments by Mr. Polinière Pierre Polinière (1671–1734) was a pioneer of experimental physics in France and a contemporary of Bélidor., who reports all its dimensions. Here are some experiments that may give an idea of how the others are performed.

Why an animal dies in the receiver when the air is expanded.

804. If a small animal is placed under the receiver, as the air is pumped out, it is seen to fall into a faint; because the air it has in its lungs and in its blood, ceasing to be in equilibrium with that which it is accustomed to breathe, expands and prevents the circulation of the blood from occurring as usual. If one continues to expand the air even further, the animal dies; and if care is taken to count the number of piston strokes given to cause its death, one can then find by calculation how much the air had to be expanded for it to cease being breathable for that animal. But it must be noted...