1st. Let AD be equal to D c No. 1. Then the angle B A C is a right angle.

Because A D is equal to DC, the angle D A C is equal (Prop. 5) to the angle DCA. For a like reason, the angle DAB is equal to the angle DBA. Therefore, the angle BAC is equal to the angles ABC and A CB (Ar. 2). But the three angles BAC, ACB and A B C, are equal (Prop. 32) to two right angles. Therefore the angle BAC (Cor. 5) is a right angle.

2nd. Let AD be greater than DC No. 2. Then the angle BAC is acute.

Because DA is greater than D C, the angle DA C is less (Prop. 18) than the angle DCA. For a like reason, the angle DA B is less than the angle DBA. Therefore the angle BAC is less than the two angles A B C and a C B. Therefore the angle B A C (Cor. 6) is acute. 3rd. Let AD be less than D C No. 3. Then the angle BAC is obtuse.

Because DA is less than D C, the angle DAC is greater than the angle DCA (Prop. 18). For a like reason, the angle DA B is greater than the angle DBA. Therefore the angle BAC is greater than the angles A B C and AC B. Therefore the angle BAC (Cor. 6) is obtuse. Wherefore, any angle of a triangle is right, acute, or obtuse, etc. Q. E. D.

EXERCISE IV. TO PROPOSITION XXXII.*

The straight line drawn from the vertex of any angle of a triangle bisecting the opposite side, is equal to, greater than, or less than half that side, according as the angle is right, acute, or obtuse. In fig. h, let A B c be a triangle, and let A D be drawn from a bisecting B C in D.

1st. Let the angle B A C No. 1 be a right angle. Then A D is equal to D C. For, if not, then A D is either greater or less than D c; and, according as A D is greater or less than D c, so is the angle BA c acute or obtuse, by the preceding exercise; but it is neither acute nor obtuse by hypothesis; therefore A D is neither greater nor less than D c-that is, A D is equal

to D c.

2nd. Let the angle B A C No. 2 be acute. Then A D is greater than D c. For, if not, then A D is either equal to, or less than, DC; and, according as A D is equal to, or less than, DC, so is the angle B A C right or obtuse, by the preceding exercise; but it is neither right nor obtuse by hypothesis; therefore A D is neither equal to, nor less than, D C-that is, A D is greater than D C.

3rd. Let the angle BAC No. 3 be obtuse. Then, it may be shown, as in the preceding cases, that A c is less than a D. Wherefore the straight line drawn, etc. Q. E. D.

EXERCISE V. TO PROPOSITION XXXII.†

If the sides of an equilateral and equiangular pentagon (or fivesided figure) be produced till they meet, the angles formed at the points of meeting are together equal to two right angles.

sides A B, BC, CD, DE and EA, both ways till they meet in the points F, G, H, K and L. Then all the angles at these points are together equal to two right angles.

Because the angle FA E is the exterior angle of the triangle LA H, it is equal to the two interior and opposite angles at H and L (Prop. 32); and for a like reason, the angle A EF is equal to the angles at G and K; therefore (Ax. 2) the two angles FAE and AE F are equal to the four angles at H, L, G and K. To these equals add the angle at F; and the three angles FAE, EA F and A FE are equal to the five angles at the points F, G, H, K and L. But the three angles FEA, EAF and AFE are equal to two right angles (Prop. 32); therefore (Ax. 1) the five angles at the points F, G, H, K and L, are equal to two right angles. Wherefore, if the sides of an equilateral, etc. Q. E. D.

EXERCISE VI. TO PROPOSITION XXXII.

If the sides of an equilateral and equiangular hexagon (or six-sided figure) be produced till they meet, the angles formed at the points of meeting are together equal to four right angles.

In fig. k, let A B C DEF be a regular hexagon. Produce the sides AB, B D, DC, CE and EF, both ways till they meet in the points G, H, K, L, M and N. Then all the angles at these points are equal to four right angles.

Walker and Smith arises from the fact that the former denotes accent, the latter quantity. Walker should be followed in pronunciation.

D. D. N. (Slough): Pronounce o, n, To, as hee, he, toe, ny as seen, and wv as tone. It is unnecessary to learn the English-Greek exercises. The exercises are well done-particularly the English-Greek. Instead of "The king he is a general," you should have said "The king himself," etc.

H. T. S. H. (Oxford): We fear that his scheme for an air-pump, however ingenious, would not answer; we strongly doubt the applicability of glass pistons.-OMEGA: We must try and insert the biography he wants in the P. E.-J. Mc BAIN (Fetter-lane): The Cotton Account is not debited for the amount of cash received, but for the amount due by the merchant for the cotton purchased on credit or otherwise; and the same account is credited by the amount due for the cotton sold. The reason why the amounts due are entered, less the discounts, is because the latter are small and customary, and not worth opening a separate account for; besides, the sums, less the discounts, are the actual sums for which the cotton was purchased or sold, and are, after all, the proper sums to be entered in the Ledger. The incidental expenses arising from the sending of the cotton to a distance are, in like manner, added to the cost of the article, as constituting the real amount due for it, or the actual sums for which the cotton was purchased or sold. As to Petty Cash, its name implies that it has nothing to do with Discounts, Incidental Expenses attending the purchase or sale of goods, Interest, or any sums of money laid out in trade transactions; it is only applicable to small disbursements which occur in the counting-house, for stationary, postages, coals, candles, etc.-LABORE VINCO (Soho): His solutions of the Centenary Problems are very good, but they have come to us so irregularly and by piece-meal, that we could not insert their number

This exercise was solved by J. H. EASTWOOD (Middleton); T. Bocook (Great Warley); E. J. BREMNER (Carlisle); and WARIN (East Dereham)

ON PHYSICS, OR NATURAL PHILOSOPHY.

No. XXXV.

(Continued from page 110.)

EXPANSION AND DENSITY OF THE GASES. Laws of Expansion.-M. Gay-Lussac first proposed the two following laws on the expansion of gas, which have been, till very recently, considered as correct, and universally received. 1st, That all gases have the same co-efficient of expansion, that is, the same increase in volume in passing from 0° to 10 Centigrade, and that its numerical value is 0.00375 or 2663 Bay 27.

1

2nd. That this co-efficient is independent of the pressure, that is, of the original density of the gas. These laws are founded on the supposition that all gases expand equally under an equal increase of temperature, and that the expansion is in direct proportion to the temperature. The method adopted by Gay-Lussac in the determination of these laws was the following: the expansion of air and other gases was measured by an air-thermometer formed of a spherical bulb A, and a capillary stem AB, fig. 183.

The part of the stem between B and C was divided into parts of equal capacity; and the number of these parts which the bulb A contained, was ascertained by weighing the apparatus when full of mercury at 0° Centigrade, and then heating it gently in order to force out of it a little of the mercury. By weighing it again, the weight of the mercury forced out was determined. By cooling the remainder down to 0 Centigrade, a vacuum was produced which showed the volume of the corresponding weight of the mercury which was forced out. From this, the volume of the mer

diminished in volume, and the index at в was forced towards the bulb at A. The point where it became stationary was marked, and this determined the volume of the air at 0° Centigrade, since the capacity of the bulb was known. The ice was then removed and replaced by water or oil, and the box was heated over a furnace. The air in the bulb expanded, and the index advanced from A towards B. The point where it became stationary was then marked, and at the same time the temperature indicated by the two thermometers D and E was noted, so that the volume of air and its temperature were both ascertained.

Now if in the first instance we suppose that the atmospheric pressure did not vary during the experiment, and neglect the expansion of the glass, which is very small, the total expansion of the air in the apparatus will be found by subtracting the volume which it had at 0° Centigrade from the volume which it had at the end of the experiment. Then, by dividing the the expansion corresponding to 1° Centigrade will be deterremainder by the number of degrees in the final temperature, units contained in the volume at 0° Centigrade, we obtain the mined; and again dividing this quotient by the number of expansion corresponding to one degree and one unit of volume, that is, the co-efficient of expansion. In the following problems, the corrections for the variations in the pressure of the atmosphere and the expansion of the glass are taken into

account.

Problems on the Expansion of the Gases.-1st. Given v the volume of a gas at 0° Centigrade, and the co-efficient of expansion a, what will be its volume v' at the temperature to Centigrade, the pressure remaining constant. Here, the same reasoning being employed as in the case of linear expansion given in a former lesson, we shall, at once, have v'v+avt; whence v=v (1+at) (1.)

2nd. Given v' the volume of a gas at the temperature to, and the co-efficient of expansion a, what will be its volume v at 0° Centigrade, the pressure remaining constant? Here, from

cury remaining in the apparatus was deduced, and consequently the volume of the bulb, by proportion, as already shown in the case of the piesometer, p. 105, vol. iv. The bulb and the stem were now filled with dry air, in the following manner: they were first filled with mercury, which was boiled in the bulb in order to dry it; a tube c, filled with drying substances, such as chloride of calcium, was then fixed to the extremity of the stem by means of a cork. Into the stem AB, through the tube, was next introduced a fine platinum wire, by means of which the interior of the tube was agitated; and it was at the same time held in an inclined position, so as to permit the mercury to flow out drop by drop, by slightly shaking the apparatus. The air then entered the bulb A, bubble by bubble, after having been dried by passing through the chloride of calcium. Lastly, there was preserved in the stem A B a small portion of mercury to serve as an index, as shown at B.

The air-thermometer was then placed in a rectangular box made of tin. This box was at first filled with ice, the air was VOL. V.

known, we have the height h of the column a c. The point b is then closed with a little wax, and the barometric pressure is noted. Now, if this pressure be denoted by H', the pressure in the reservoir B, is denoted by H'-h. The reservoir is now withdrawn from the ice and weighed, in order to ascertain the weight of the mercury which has been introduced into it. This reservoir is then completely filled with mercury at 0° Centigrade, and the weight P' of the mercury contained both in the reservoir and the tube is ascertained. Now, if k denoted the co-efficient of the expansion of glass, a that of air, and D the density of mercury at 0° Centigrade, we find a by the following process. The volume of the reservoir and of the P' tube at 0° Centigrade is from the formula PVD formerly given; consequently, at to Centigrade this volume, by a pre

(PP) (1a) (H' H)

h D

(2.) But the volumes represented by the formulæ (1.) and (2.) are the volume of the reservoir and the tube at the pressure ; they are therefore equal. Whence, by suppressing the common denominator, we have the equation

P' (1 + kt) H = (r′ —P) (1 + a t) (í — h) ; (3.) from which may be deduced the value of a.

By experiments and calculations of this kind, M. Regnault found that from 0 to 100 Centigrade, and for barometric pressures between 11.8 inches and 59.1 inches, the co-efficients of expansion for certain gases were as follows:

stationary. Hence the indications of the air-thermometer are influenced by the pressure of the atmosphere, and thus require to be corrected at every observation.

When variations of temperature of considerable extent are to be measured, the air-thermometer is furnished with a tube similar to that employed in measuring the co-efficient of the expansion of gases in the apparatus of M. Regnault (figs. 184 and 185). Making experiments with this tube, as explained in the description of that apparatus, the quantities P, P', H, H', and h, which enter into equation (3.) are determined; and as a is known, this equation, by reduction, will give the temperature t, to which the tube has been raised. According to the researches of M. Regnault, the air-thermometer sensibly agrees with the mercurial thermometer as far as 260° Centigrade, or 500° Fahrenheit; but beyond this point the mercury expands more rapidly than the air.

Density of Gases.-The specific weight or density of a gas is the ratio of the weight of a certain volume of the gas to that of the same volume of air, the gas and the air being both at the same temperature and pressure; and in the use of the Centigrade thermometer, this temperature and pressure are 0° Centigrade and 29.922 inches, which we shall call the standard temperature and pressure. According to this definition, the density of a gas is found by determining the weight of a certain volume of the gas at the standard temperature and pressure; next, the weight of the same volume of air at the standard temperature and pressure; and then by dividing the former weight by the latter. In determining the density of gases, a glass vessel or globe, whose capacity is about that of two imperial gallons, is employed, having a stop-cock at the neck or aperture, which can be screwed to an air-pump. This globe is weighed first when empty, that is, when a vacuum has been made in it; second, when full of air; and third, when full of the gas whose density is to be found. The air and the gas are dried by the same process as that followed in the experiments made with the apparatus represented in fig. 184. By subtracting the weight of the globe when empty from its weights respectively when full of air and full of the given gas, we have the weights of the same volume of the air and of the gas. In the case where the experiment has been made at the standard temperature and pressure, it is sufficient to divide the weight of the gas by the weight of the air, and the quotient will be the density required. The process of finding the density of a gas in general requires numerous corrections, in order to refer it to the standard temperature and pressure, as well as to reduce the temperature of the glass vessel to 0° Centigrade. These

These numbers show that the co-efficients of the expansion of gases differ only by quantities which are extremely small. M. Regnault has proved, also, that at the same temperature the expansion of any gas is greater in proportion to the increase of pressure; and that the co-efficients of the expansion of two gases differ more when they are subjected to greater pressures. The Air Thermometer.-This thermometer, as its name indicates, is founded on the principle of the expansion of air. When it is intended to measure small variations of temperature, the same form is given to it as that employed by Gay-Lussac in order to measure the co-efficient of the expansion of the gases (fig. 183); that is, it is composed of a glass bulb, to which is cemented a long capillary tube or stem. The bulb is filled with air perfectly dry; and there is introduced into the stem, a small quantity of sulphuric acid, coloured red, to serve as an index; the instrument is then graduated by comparing its indications with those of a mercurial thermometer. The extremity of the stem of this thermometer must be allowed to remain open; otherwise, the contraction or expansion of the air above the index would take place at the same time as that of the air in the bulb, and the index would remain

given in this lesson for the solution of problems of this description, but most of these may be avoided by the following method:

M. Regnault has applied to the preceding process some modifications, which dispense with part of the corrections. For this purpose, the globe, which is employed to weigh the given gas, is hung on the one scale of a balance, and it is brought into equilibrium with another globe of the same volume herThese two metically sealed and hung on the other scale. globes expanding equally under the same degree of temperature, always displace the same quantity of air, whence the variations in the temperature and pressure of the atmosphere have no influence on their weights. Now, when the first globe is successively filled with the air and with the gas whose density is required, it is placed in a zinc vessel, and surrounded with ice. By this means it is brought to the standard temperature; and by shutting the stop-cock when the gas introduced into it is at the same temperature, the corrections for temperature are avoided. Lastly, the two gases can be easily brought to the standard pressure by referring the pressure under which the experiment is made to the law that the weights are proportional to the pressures.

In the case of gases which act upon brass, as chlorine for example, a brass stop-cock cannot be employed. It is then necessary to employ a glass bottle with a ground stopper, and to introduce the gas by a bent tube which reaches to the bottom; the bottle being held upright or inverted, according as the gas introduced into it is heavier or lighter than the air. When all the air is expelled from the bottle, the tube is removed, and it is closed by the stopper. If the bottle be then weighed full of gas, the weight obtained will include the weight