Chap. VIII.

GEOMETRY.

827

Fig. 1028.

multiply the distance between the two stations by 12, and divide this product by theremainder from the subtraction of the two quotients; the quotient of this last divisionadded to the height of the staff gives the required height.

For measuring inaccessible Heights with an Astrolabe , as that of A B, select two stations asC and D, distant from each other in this case 36 feet; suspend the astrolabe at the stationC, look at the point A throughthe sights, and observe at whatpoint on the altimetric scale thealhidade stands, as at the 6point on the versed shadow.

Perform a similar operation atthe station D, that is to say, ob-serve the point A through thesight of the alhidade, noting atwhat point it stands, as at thethird point of the versed shadow,which shows that the first of thepreceding observations must be

followed; divide 12 by 6, the number of divisions which the alhidade indicated on theversed shadow at the first station, and the quotient will give 2: then divide 12 by 3,the number indicated by the alhidade on the side versed shadow of the second station,the quotient will give 4; subtract these two quotients from each other, as 4 — 2 = 2, withwhich divide the number of feet distant between the stations C, D, 36, which leaves 18to be added to the height of the staff 5, making 23 feet for the required height, A B.

Or if it is required to measure the height AB, in another ease, choose the two stationsC and D, which here are 27^ feet apart; then suspend the astrolabe at the station C ;by the staff’and by the sights of the alhidade bonethe point A, and mark where the alhidade standson the altimetric scale, which at the first stationwill be at the right angle figure 12, where theright and the versed shadows meet: suspend theastrolabe at the second station, and bone the lineA through the alhidade, remarking where it standson the altimetric scale, as at the 8 point in the versedshadow, in which case we must, to finish theoperation, employ the second of the previous generalrules: therefore, multiply the distance between thetwo stations C and D, which is 27£ feet, by the

number of the side versed shadow at which thealhidade stood, viz. 8x27^ = 220 + 4, the numberof points necessary to make up 12 = 55 + 5, the Fig. 1029.height of the staff, and we have 60 feet for therequired height A B, which is found without approaching theline.

Another method to find the height PR may be described,which is an example coming under the third system. Take astation at any point S, and having planted the staff per-pendicularly, suspend the astrolabe from the left hand, elevatingor depressing it until the point It is seen from the top of thestaff, and by looking through the sights of the alhidades:observe at what point on the altimetric scale the alhidadestands, as at the 4th point of the right shadow, then, accordingto the third rule, we have 12 + 4 = 3; then retire in a straightline to the station S, and plant the staff perpendicular to, and ona level with the point P, suspending the astrolabe also, until thepoint R is seen, by looking along the staff T, and through thesights of alhidade: after having observed where the alhidadestands, as at 10 on the versed shadow, this number 10 will serveto divide 12, the number of parts on the staff T; the quotientwill be 1, which subtracted from 3, the quotient of the firstdivision, there remains 2; this number 2 denotes that theheight R P is twice as great as the distance between thestation S and T; that is to say, if the distance S T is 8 feet,the height from P to R will be 16 + 5, when the height ofthe staff is added: if the difference had been 1 only, then thedistance between the two stations with the height of the staffwould be equal to the required height. Lastly, if the difference

Fig. 1030.