Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopædia Britannica/Aræometer

by William Archibald Cadell

History. ARÆOMETER (composed of αραιος, levis, tenuis, and μετρον mensura), a measure of the comparative density and rarity of bodies. The name does not occur in ancient authors; hydroscopium and baryllium being the ancient names of the instrument. This instrument was known in the civilized part of the Roman empire, about the year 400, as appears from the fifteenth epistle of Synesius, addressed to Hypatia, daughter of Theon; and to Hypatia some modern writers have erroneously ascribed its invention. The instrument is also described in some verses annexed to Priscian; and the principles on which its operation is founded, are to be seen in the treatise of Archimedes on floating bodies (De Humido Insidentibus). The term, as used by writers on natural philosophy, is chiefly applied to instruments which are made to float, so as to indicate the specific gravity of the liquids in which they are placed: the pêse liqueur, and hydrometer, in common use for measuring the specific gravity of vinous spirits, are instruments of this kind.

Theory. A floating body displaces a portion of the liquid, the weight of which is equal to its own weight, the liquid acting upwards with a force equal to this weight, and the weight of the body acting downwards with the same force, equilibrium takes place. If the body be afterwards placed in a liquid of less density, the part of the body immersed, will be greater than when the body was in the more dense liquid, because it requires a greater volume of this less dense liquid to equal the weight of the floating body. The absolute weights of two bodies being the same, their specific gravities are in the inverse ratio of their volumes ${\frac {G}{g}}={\frac {v}{V}}$, when $G$ is put for the specific gravity of the first body, $g$ for that of the second; $V$ for the volume of the first, and $v$ for the volume of the second. Aræometer with Scale.On this principle the common hydrometer is constructed; the instrument described by Synesius, is also of this kind. In order that a small difference in the volume immersed may be sensible, the part which is intersected by the surface of the fluid is in the form of a very slender cylinder, the great bulk of the instrument being always

immersed in the liquid. At the inferior part is a small ball, containing mercury or small load shot, which serves as ballast, bringing the centre of gravity low, so that the instrument may float erect, and without much lateral oscillation. The common hydrometers are made of glass, and sometimes of brass, or tin or pewter, and some have been made of amber as objects of curiosity: when made of glass, a scale, inscribed upon paper, is inserted in the cylindrical stalk; the division of the scale at which the surface of any liquid intersects the stalk, denotes the specific gravity of that liquid. Mode of forming the Scale.The divisions of the scale should be formed by immersing the instrument in liquids of known specific gravity, and marking a number corresponding to that specific gravity opposite to each division. The specific gravities of water and alcohol mixed in various proportions, have been accurately ascertained by Mr Gilpin (see his Tables, and Dr Blagden’s paper in the Philosophical Transactions); on immersing the instrument in a mixture of known proportions of these two liquids, the point at which the surface intersects the stalk, is to be marked with the number expressing the specific gravity of the mixture taken from the table. Some hydrometers, such as that constructed by the French chemist Beaumè, and which is much used in France under the name of Aréometre de Beaumé, have the scale divided into equal parts, so that the divisions do not correspond as they ought to do with the numbers which express specific gravities.

Fahrenheit’s. In the aræometer of Fahrenheit, the uncertainty arising from the erroneous division of the scale is obviated, no division being required. The form of the instrument is the same as that just described,

only at the top there is a small cup, into which weights are put, so as to bring the surface of the denser liquid to a fixed mark on the stalk; when the instrument is placed in a liquid of jess density, some of the weights are taken out till the mark again comes to the surface. Manner of using it. Suppose the weight of the instrument and of the weights in the cup together equal to 1000, when sunk to the mark in distilled water at a certain temperature; the instrument is now taken out of the water and immersed in a liquid, where 10 must be taken out of the cup in order to bring the mark to the surface; the immersion in water indicates that a volume of water weighs 1000; the immersion in the second liquid, shows that an equal volume of this liquid weighs 990; when the volumes of bodies are equal, the specific gravities are directly as the absolute weights ${\frac {G}{g}}={\frac {W}{w}}$, consequently the specific gravity of the second liquid is 990, that of water being 1000. Construction of the Weights.To save computation, it is convenient that the whole weight of the apparatus, when in distilled water, at a certain temperature, should be represented by 1000; for this purpose, the instrument-maker divides the weight of the apparatus into 1000 parts, and forms small weights consisting of one, two, three, &c. of these thousandth parts, the relation of which to the ounce or pound, does not require to be known; the weights thus formed are to be used with the instrument.

Nicholson’s. The aræometer of Nicholson is like that of Fahrenheit, with the addition of an immersed cup, whereby it is rendered proper for ascertaining

the specific gravity of solids. Mode of using it.Suppose that it requires 400 grains in the exterior cup to sink the instrument to the mark in distilled water, at 60 degrees of Fahrenheit’s thermometer; 1st, The body under examination is put into the exterior cup, and weights (say 300 grains) are taken out till the mark again stands at the surface; this gives the absolute weight of the body 300 grains. 2dly, The body is then put into the immersed cup S, taking care to brush off any air-bubbles with a hair pencil, and in order to bring the mark to the surface, a weight (say 100 grains) must be put into the exterior cup, that is, the weight of a volume of water equal to the body, is 100 grains. The first part of the process gave the absolute weight of the body 300 grains, and the volumes being equal, the specific gravities are as the absolute weights, consequently the specific gravity of the body is 300, that of water being 100. This aræometer may be used to find the specific gravity of liquids; the process, in that case, is the same as that described above in speaking of the aræometer of Fahrenheit. The aræometer of Nicholson is useful to the mineralogist for ascertaining the specific gravity of minerals; the specific gravity being a convenient character for distinguishing one kind of mineral from another. It is sometimes made of tinned iron, but where more accuracy is required, copper is the material employed. When put together, it does not exceed a foot in length, and therefore is suited to form a part of the travelling mineralogist’s apparatus.

Some aræometers have been constructed with the

exterior cup C placed underneath, and supported by a stirrup, whose upper part is fixed to the stalk of the aræometer, as represented on the margin; this is done in order to place the centre of gravity low, that the aræometer may thereby float mere steadily. The aræometer floats in a cylindrical vessel fitted to the size of the stirrup, and this vessel is supported on a stand so formed as not to interfere with the free motion of the stirrup.

Deparcieux’s. The aræometer of Deparcieux is like the common hydrometer, only the ball is much more voluminous; this renders it capable of indicating the small difference which exists in the specific gravity of the water of different springs, for which purpose Deparcieux proposed it. The dilatation of the large glass bulb by heat, has a considerable effect on the operation of this instrument, and this dilatation being differont in different instruments, renders the results inaccurate. The different aræometers above-mentioned, have the advantages of being easily made and easily carried about; but where the specific gravity of a body is required with the greatest accuracy, recourse must be had to the hydrostatic balance, which ought to be constructed with the utmost care by the most skilful artist.

Formula. The following algebraic expressions may serve to elucidate some of the properties of the aræometers hitherto spoken of:

$g$ is the specific gravity of water, which is 1000 ounces when the ounce and foot are taken as unities, 1000 ounces avoirdupois being the weight of a cubic foot of water.

$z$ is the diameter of the wire-stalk of the aræometer.

$\pi$ is 3.1415, &c. the number expressing the periphery of a circle whose diameter is 1.

${\tfrac {1}{4}}\pi z^{2}$ is the surface of a transverse section of the wire-stalk.

$v$ is the volume of the bulb or body of the aræometer.

$w$ is the whole weight of the aræometer.

$x$ is the length of the stalk that is plunged in the water.

${\tfrac {1}{4}}x\pi z^{2}$ is the volume of the immersed portion of the stalk.

When the aræometer floats in equilibrio, it displaces a volume of water equal to its own weight, therefore, $w=g(v+{\tfrac {1}{4}}x\pi z^{2})$, and, $g={\frac {w}{v+{\tfrac {1}{4}}x\pi z^{2}}}$, $x={\frac {4(w-gv)}{2g\pi z^{2}}}$; $w-gv$ is the difference between the quantity of water displaced by the whole aræometer, and the quantity displaced by the bulb alone, $w-gv$, therefore, is the volume af water displaced by the immersed portion of the stalk, as the diameter of the stalk $z$ is very small, the cylinder of water $w-gv$, which has $z$ for its diameter, is likewise very small, and does not exceed a few grains in weight; therefore, a small variation in $w$ (the weight of the aræometer), or in $g$ (the density of the liquid), occasions a great variation in $x$ (the length of the immersed part of the stalk). The value of $x$ changes rapidly, when $z$ (the diameter of the stalk) is changed, because the value of $x$ is divided by $z^{2}$, which is the square of a very small quantity.

Sensibility to the sp. gr. of Liquid. When the aræometer is immersed in a liquid of another specific gravity $g^{1}$, then the equation is $x^{1}={\frac {2(w-g^{1}v)}{g^{1}\pi z^{2}}}$; subtract the value of $x^{1}$ from that of $x$, and there results $x-x^{1}={\frac {2w(g-g^{1})}{gg^{1}\pi z^{2}}}$; this is the diminution in the length of the immersed part of the stalk, which takes place when the aræometer is transferred to a liquid of a greater density. By this formula, it is seen, that the sensibility of the aræometer, that is, the length of the portion of the stalk which emerges upon transferring the aræometer to a denser liquid, is augmented, in the first place, by increasing $w$ (the weight of water displaced by the aræometer), that is, by increasing the volume of the body of the aræometer; secondly, by diminishing $z$ (the diameter of the stalk), which is in the denominator of the value of $x-x^{1}$. Consequently, the faculty of the aræometer to show the different densities of liquids is, in general, expressed by the fraction ${\frac {w}{z^{2}}}$.

Expression of the Sensibility to additional Weight. With regard to the vertical mobility of the aræometer, when put in motion by placing a small weight ($s$) in its exterior cup, substitute $w+s$ for $w$, then, $x^{1}={\frac {4(w+s-gv)}{2g\pi z^{2}}}$, take the difference between this and $x={\frac {4(w-gv)}{2g\pi z^{2}}}$; this difference is $x^{1}-x={\frac {4s}{g\pi z^{2}}}$. Which shows that the length of the portion of the stalk that a small weight causes to immerge, is proportional to ${\frac {s}{z^{2}}}$, or in the direct ratio of the small weight, and in the inverse ratio of the square of the diameter of the stalk.

Expression of the Diameter of the Stalk. When the small weight, the density of the liquid, and the length of that part of the stalk which is submerged on adding the small weight, are known, then this equation will give the diameter of the stalk in known quantities $z=2{\sqrt {\frac {s}{g\pi (x^{1}-x)}}}$.

Density of a liq. expr. in terms of $w$. When the weight of the whole aræometer is known in ounces, &c., and the specific gravity of one of two liquids (water for instance) is known, the difference of specific gravity between that liquid and another liquid may be had in known quantities.

$g$ is the specific gravity of water.

$g^{1}$ is the specific gravity of the second liquid, which is here supposed more dense.

$w$ is the weight of the volume of water displaced by the aræometer.

$s$ is a small additional weight placed on the exterior cup to keep the aræometer, when placed in the denser liquid, at the same point of immersion as when it floated in water.

$w+s$ is the whole weight of the apparatus when floating in the denser liquid. ws

The equation $g^{1}={\frac {w+s}{v+{\tfrac {1}{4}}x\pi z^{2}}}$ is obtained by substituting $g^{1}$ for $g$, and $w+s$ for $w$ in the equation, $g={\frac {w}{v+{\tfrac {1}{4}}x\pi z^{2}}}$, which was given above. Divide by $g={\frac {w}{v+{\tfrac {1}{4}}x\pi z^{2}}}$, and there results ${\frac {g^{1}}{g}}={\frac {w+s}{w}}$, which gives the proportion of the density of the second Iiquid to the density of water. By subtraction there results ${\frac {g^{1}-g}{g}}={\frac {s}{w}}$ and $g^{1}-g={\frac {sg}{w}}$, that is, the difference between the density of the second liquid, and the density of water is found by multiplying the small weight by 1000 ounces, and dividing this product by the number of ounces, &c., which denote the weight of the aræometer uncharged.

Bead Aræometer. Small bodies, whose specific gravities are known, serve to indicate the specific gravity of a liquid in which they just remain suspended. In this way, beads of glass, three or four tenths of an inch in diameter, are employed, each of which remains suspended in spirit of a certain specific gravity. The density of each of these beads, or rather bubbles, is regulated by the proportion between the quantity of glass and the cavity which the glass incloses. A piece of boes-wax, whose specific gravity, by the addition of lead, is such, that the body is just suspended in brine of a known density, is used as an aræometer In some salt works. The fresh egg of a common fowl is just sustained by brine of a certain specific gravity, and is employed as an aræometer.

Homberg’s. The aræometer of Homberg, consists of a phial, with a slender neck and glass-stepper, so made, that it may be filled with the same volume of different liquids. It is employed in finding the specific gravity of liquids in the following way: 1st, The phial is filled with distilled water, and then weighed in a balance; 2dly, The phial is emptied, and again filled with the liquid, whose specific gravity is sought, and weighed in a balance, the proportion of the weight of the contents of the phial in the second process to the weight of its contents in the first, is the specific gravity required. The inconveniences which have prevented this method from being generally used, are; the difficulty of completely cleaning the phial from the liquid which it previously contained; the difficulty of filling the phial exactly with the same volume of each liquid; and the variation of the volume of the phial from changes of temperature.

Pump Aræometer. The pressure of the atmosphere supports columns

of different fluids, whose height is inversely as the densities of the fluids. An aræometer has been constructed on this principle. It is a curved tube, one leg of which has its extremity immersed in water, and the other in the spirit whose density is to be tried. On rarifying the air in the tube, by means of a pump fixed at the upper part of the tube, the water ascends in one leg, and the spirit in the other; the height of the column of each liquid being measured by a scale of equal parts applied to each branch of the tube. This instrument has never come into use, probably on account of the difficulty of ascertaining, with precision, the points at which the surfaces of the columns are terminated. See Encyclopædia, Art. Hydrodynamics, Part I. ch. 2. sect. 2. for some farther notice of Aræometers. (Y.)