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distance+c, c being the distance between the two stations of the shorter staff ED; also the lengths of the staves remaining still the same, ED will be represented by d, as above.

Now, from the preceding proportion we shall have, by substituting for ID, ED, IA, and AB, the above letters: 1st operation 2d operation Whence, by

a :d= x

plane of the figure's base, as at C, and when the object is accessible, measure the distance CA. Now retire back in the direction AC to D, till the eye observes the top of the object exactly in the centre of the mirror, which, for the greater degree of accuracy, may be marked by a line across it. Then having measured the disy tance DC, and ascertained the :d=x+cy height of the eye of the observer, it will be, from the known laws of subtraction, ala: d = c reflection, as DE X CA DC: DE = CA: AB, DC the altitude of the object required. When the object is inaccessible, that is, when the distance CA cannot be measured, two such operations as that above must be employed. Thus,

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that is, y = AB=

dc alsa

: y

the height

of the tower required.
For the method of determining
the distance IA, see the conclu-
sion of problein 4.

To measure accessible or inaccessible Altitudes by means of Shadows. (Plate I. fig. 2.)

At any time when the sun shines, plant a rod ab perpendicularly at a, and measure the length of its shadow, and, immediaately after,

measure the shadow of the

pro

posed object AB. Then by similar triangles,

ba X CA
ca

AB,

ca: ba CA: the altitude required. If the object be inaccessible, but still such that the difference of the lengths of its shadows, taken at two different times, can be ascertained, the altitude may be found nearly the same as in the last example. Thus, make ab=a, ac=b, and the unknown length of the shadow; let the second shadow of the rod b', and the second shadow of the tower ad, the height of the towery; then, by the preceding proportion,

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=

Let ED=a, DC=d, and the unknown distance CA; also let d' represent the analogous distance D/C in the second operation, and ac the second distance C/A, that is, making c the distances between the two situations of the mirror; and let the required height of the object = y. Then substituting the above letters in the preceding proportion, we have, 1st operation d 2d operation d' Whence, bydd ; a= subtraction,

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: y a = x + c: y

and consequently, y=
altitude sought.

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:y

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To measure an accessible or inaccessible Object, by the Geometrical Square. (Plate I. fig. 4.)

Having fixed the instrument at the centre of motion D, till the any place C, turn the square about top of the object B is perceived in y the direction of the sights placed a=x+dy on the side of the square DE, and note the number of divisions; cut off the other side by the plumbline EG; then, having measured the distance CA, we have by similar triangles, as

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br the altitude

To measure accessible or inaccessible Objects by means of Optical Reflection. (Plate I. fig. 3.) Place a mirror, or other reflect ing surface, horizontally in the

EF FG CA or DH:
CA X FG

EF

= BH;

to which adding the height of the observer's eye, DC, we shall have

CA X FG EF tude sought.

+ DC AB, the alti-with a quadrant, theodolite, or At any convenient station C' other graduated instrument, measure the angle of elevation ACB; then, having also measured the distance CA, we have, from the elementary principles of trigonometry, as

;

In the case of an inaccessible object, two observations must be made similar to that above, in which the only variable lines will be GF and CA. Let, therefore, the side of the squares, and the variable part GF a, in the first observation, and al in the second also put the unknown distance in the first case, and xc, in the second, so that c will be the distance of the observer's two stations, and make the required height of the objecty. Then on the same principle as those above, 1st operation s: α = 2 2d operation s : a Whence, by equality,

axalxalc, or x=

but by the first y =

therefore y=

:y

= x + cy

alc as d

ax

S

aal c s(as al the altitude required.

Rad. AC tan. ACB: AB, the altitude required.

Or by logarithms, log. AB=
log. AC+ log. tan. ACB-log. rad.
Suppose, for example, the dis-
tance AC340 feet, and the angle
of elevation ACB = 34' 30; then,
by the above formula,

Log. AC or log. 340

2.5314789 Log. tan. 34' 30. .9-8371343

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It is obvious, that the methods made use of in all these problems, for inaccessible objects, will give the distance of the objects as well as their altitudes. Thus, Prob.

1. The distance

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To measure an inaccessible Ob ject by two Stations. (Plate I. fig. 6.) Let AB be the object of which the altitude is required.

the two stations D and C, and mea
Take the angles of elevation at
sure the distance between them.
The angle DBC= the difference
of the two measured angles RCA
and BDA, (Euc. prop. 16, book 1.),
which angle therefore becomes
known. And by trigonometry, as
sin. DBC: DC sin. CDB:
DC X sin. CDB

sin. DBC

= CB.

Again, as radius: CB sin. ACB :
CB X sin. ACB

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: BA,

the altitude required.

Or substituting, in the second expression, the value of CB, we have

BA =

DC. sin. CDB. sin. ACB.

Rad. sin. DBC.

Or by Logarithms.-From the sum of the logarithms of the terms

It is singular that the above methods for inaccessible altitudes and distance, being so very sim-in the numerator subtract the sum ple, should never have been me tioned by any author; at least they are not mentioned by any one that we are acquainted with.

To measure an accessible Object Trigonometrically. (Plate I. fig. 5.) Let AB be the object of which the altitude is required.

of those in the denominator, and the remainder will be the loga. rithm of the required altitude BA.

To measure Altitudes by the Barometer.-The application of this instrument to the measuring of al titudes, suggested itself to mathe. maticians, soon after the experi

ment of the Puy de Domme, which | progression; hence, calling the denwas executed in order to confirm sity at the surface equal to d", and the Torricellian experiment. the several altitudes 1, 2, 3, &c. we shall have

Alt. 0, 1, 2, 3, 4.

cors.

der. dn, an—1, an—2, dn—8, dn—4,

&c.

Or, dividing the latter series by d",

2, 3, 4, &c.

Let EAQ, (Plate I. fig. 7.) represent the surface of the earth, and A, B, C, D, &c. a column of the atmosphere. Conceive this to be divided into a number of equal and indefinitely smail parts, AB, BC, CD, &c. in each of which we we have may suppose the density to be uni-. Alt. 0, 1, form, because they are indefinitely small. Now since the density of recip. the air is always directly as the dens. 1, d1, d2, d—3, d—4, &c. compressing force; and the com- which is strictly analogous to the pressing force being in this case property of logarithms. In fact, the pressure of the superior strata, the several altitudes form a pecuwe shall have the density of the liar system of logarithms, of which air in any of those parts, as the the reciprocals of the correspondweight of the column of atmos-ing densities are the natural numphere above; that is, if P represent generally the pressure, and D the density at any place, and Pand D/ the pressure and density at any other place whatever, we shall have

bers, and which may, therefore, properly enough, be denominated atmospheric logarithms, on the same ground as the Napierian are termed hyperbolic logarithms. We shall consider them in this way, P: D= P: D/; and shall denote them by λy, to and, consequently, the pressure is aistinguish them from the common to the density in a constant ratio, logarithms, which are generally which may be represented byn: 1;.arked log.; and from the hyperthat is, P: DP: D'=n: 1; bolic, which are denoted by h. log. hence, D= P, D' = P', &c. Hence, if u and A represent any two altitudes, and d and D the or, the density at any place is, th corresponding densities, we shall A== of the pressure of the superior λογ. d strata at that place. α = - λογ. D

n

have

If we make P represent the whence A-α = pressure at the surface A, and call each of the parts AB, BC, CD, &e.

n-1

n

P will be the

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λογ. D

d

= λόγο

D'

equal to 1, then will also Pre-Now it is well-known, that all the present the weight or pressure of various systems of logarithms are the part AB, and, consequently, convertible from one system to the other by a constant multiplier; and hence, in this case, we have only to determine the constant multiplier, that shall convert the P the density or commion logarithm of a number into its atmospheric logarithm. Let a denote this multiplier,

n

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P-P =

pressure at B;

and

no weight of BC.

In the same way

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And as in the present case, n is a great number, all the terms past | the first may be neglected; and since also M 43429448 in the common system, we have

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⚫43429448
26057

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ture. Then multiply the difference of the logarithms of the two heights of the barometer by 10000, and correct the result by adding or subtracting so many times its 435th part, (the proportion in which air expends for every degree of heat), as the degrees of the mean temperature are more or less than 31°; and the last number will be the altitude in fathoms.

Exam.-If the heights of the barometer at the bottom and top of a hill are 29.37 and 26.59 inches respectively, and the mean temperature 26°, what is the height? Log. 29.37 1.467904 Log. 26.59 = 1'424718 Diff. of Logs. 0.043186 Mult. by 10000

431-86

Now 31° 26° 5° temp. below 31°; therefore of 431.86 = 4.96; conseq. 431 86-4.96 426.90 fathoms, the altitude of the hill.

Later observations have shown, that this requires certain modifica60000, very nearly: tions, on account of the difference in low and elevated situations, the expansion of the column of mercury, and other circumstances.

d

A 60000 X log. feet.
D

Or, since the height of the mer-
cury in the barometer is always
proportional to the density, the
d m
fraction
m and M being

=

D M

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These, according to General Roy and Sir G. Shuckburgh, are as follow:

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Hence we have, according to the preceding principles, the fol lowing rule for measuring altitudes by the barometer, viz

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1+ (ƒ−32°) ·00243 }

diff. of logs. of the

d

heights of barom. diff. of degrees Fahrenheit's therm.

f= mean of the two temp. shown by the detached thermometer exposed for a few minutes to the open air, in the shade of the two stations.

Observe the height of the mercury at the bottom of the object to be measured, and again at its top; as also the degree of the thermometer in both these situations; and half the sum of these two last may The sign- takes place when the be accounted the mean tempera- attached thermometer is highest

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