A treatise upon the dendrometer, a new-invented instrument for the more certain and ready measurement of standing timber, by inspection only: for facilitating the practical operations of engineering, land-surveying, levelling, mine ing, &c. ... 1768

[Whittell, Thomas]. · 1768 · Historical

A treatise upon the dendrometer, a new-invented instrument for the more certain and ready measurement of standing timber, by inspection only: for…

Public-domain full text preserved in the Mountain Man Mining Library. Original source: archive.org.

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By the KING's Patent.

#: K Upon The N

Den D Rome Ter,

New- invented INSTRUMENT 5 FOR The more certain and ready Meaſurement of Standing Timber, By IxsPECTION ONLY : -

For

Facilitating the practical Operations of Engineering, Land-ſurveying, Levelling, Mineing, Sc.

AND FOR : Performing mechanically the various Caſes of Plane

Trigonometry, by a ſhort and familiar Proceſs, WITHOUT CALCULATION.

Lon D On,

Printed for the PATENTEEs: And fold by F. Newszzr, in Ludgate-Street; J. BENNX FETT. Inſtrument-maker to their Royal Highneſſes the Duke of 2

GLocesTER and Duke of CuuBERLAN D, in Crown-Court, 2

Soho; B. CoLsx, Inſtrument-maker, in Fleet-Street ; and T. WaiTTELL, in Pater-noſter-Row. '

4 By whom SusscRITIORs are received.

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To The Right Honourable. The

FARL OF MORT ON, Ge. Ge. &c.

Preſiden of the Royal Society, Ge. ren x „

Tion The

Dendrometer

Is nit humbly inſcribed,

By His Lorpsniy's much obliged,

and moſt devoted humble Servants,

Thomas Whittell, John Duncombe.

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DE FO RE we enter upon the ſub- B jedt of the following ſmeets, we can- E not ſsufficiently expreſs the grateful ſenſe we entertain of His MaJesTY's. condeſcenfion, in being graciousſly pleaſed to permit us to exemplify the utility of our In- Mrument in His RoYAL PRESENCE. Mi are highly obliged to the Right Honourable the Earl of MoRToN, and the reſt of the members of the Roy AL SOCIETY, for the very candid and favourable reception, which the invention met with from that illuſtrious body.

Our particular acknowledgments are due to tbe learned J. BEvis, M. D. Mr. CowLEx, Profeſſor of Mathematics at the Royal Academy at Woolwich ; Mr. SAMUEL CLARK, Teacher of the Mathematics in London; to

4397] CG aptam ;

PREFACE. Captain C OO K, Commander of the Endeavour ſloop of war (now upon a voyage to the South- Seas to obſerve the tranſit of Venus) and tg ſundry other ingenious Gentlemen; by whoſe kind aſſiſtance we have been enabled to make divers improvements both in the Book and

Inſtrument. "7 me ſhould with pleaſure acknowledge the many- perſonal civilities, which, in the courſe of our application, we have experienced from ſeveral of the Nobility and Gentry— eſpecially from two Noble Lords, and one- Gentleman of diſtinguiſhed eminence in point of abilities and merit. But as we are not at liberty to mention their names, we muſt leave by

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them to the higheſt gratification of benevolent the minds, the ſatisfattion reſulting from acts of cic

gooaneſs and generaſity.

Terms

Terms of Subſcription.

HE DenDROMETER (to be delivered with all poſſible expedition) completely finiſhed under the care of the Inventor, aſſiſted by. Dr. Beyis, &c. fitted to a proper theodolite, and contrived to act either ſseparately therefrom, or jointly therewith, with elegant mahogany-

caſe, &, - - - - Twenty-five Guineas,

To Gentlemen already poſſeſſed of a theodolite, the DENDROMETER, completely made and fitted thereto, - - - - Ten Guineas.

The DENDROMETER alone, to act without a theodolite, for the uſe of meaſuring ſtanding trees, with proper ſtaff, caſe, &c.

Fifteen Guineas.

The expence of procuring the King's Patent, the making of the Inſtruments, and other incidents, being much greater than was, at firſt, apprehended —the Patentees are induced to hope that the Subſcribers will have no objection to depoſite Five Guineas ia part of payment, upon receiving, a receipt under the hands of perſons, properly authorized, that the ſame ſhall be allowed upon delivery of the Inſtrument.

That Meſſrs. F. NREWBERY, J. BENN ET, B. CoLsz, and T. WRHITTELI, are properly authorized by the Patentees to receive the ſaid Five Guineas, and to give receipts for the ſame under their own hands, and No ONE ELSE.

,

Advertisement.

HE ParzxrEESs of the DenDROMETER

take the Liberty to acquaint the Public, that His Majeſty has been graciousſly pleaſed to grant them his Royal LETTERS PATENT for a Jack, upon a Conſtruction ſingularly new and uſeful. It is fo contrived, as to continue going for ſix Hours ſucceſſively, by a ſmall Weight, from the Height of ſix Feet only; and, when down, may be wound up with the greateſt Eaſe, It will turn a Joint of Meat of any Size, and is

ſo certain in its Execution, as to require no Attendance. It is alſo calculated to ſhew the Time,

2 aClack, while going. It is not apt to be out of Order, nor ſubject to the Expence and Trouble of Lines and Pullies; the many Difficulties of conveying them through different Rooms; the Panger of the ponderous Weight hanging to the Jacks in preſent Uſe; the frequent Neceſ- ſity and vaſt Difficulty of winding them up; nor the continual Charges of a Smoke Jack. The Price will be different, according to their

Size and Finiſhing. The ſame, u upon a more

enlarged Plan, is intended to ſerve (inſtead of a

Fire or other Engine) to raiſe Water out of Mines; whereby a prodigious Expence will be

ſaved to the Proprietors thereof.

N. B. An Aſſortment will be finiſhed with as much Expedition as poſſible.

Any Perſons inclined to favour the Park- TEES with their Commands, are deſired to write

(Poſt paid) to Mr. Thomas Whittell, in Paternofter- Row, London.

A

Introduction.

4 ;

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T is almoſt needleſs to mention, that the preſent method He MO of meaſuring ſtanding timber is a very defective, and erroneous;

Weisen g the many attempts, which have been made, to render it eaſy and certain. There is n0 inſtrument, we apprehend, hitherto made public, which can give the meaſurement with any degree of preciſion. The quadrant, altho of exten five uſe and application, will by no means anſwer that particular purpoſe: for, if the tree ſhould. either incline, or recline; or the ground, whence the obſervation is to be made, be irregular in the leaſt, the : B meaſurement

meaſurement muſt be vague and erroneous, To prove the truth of this aſſertion, we ſhall take the liberty to quote the following general rules, for the uſe of that inſtrument, from the learned and ingenious Doctor Harris: vg. On the quadrant % you have a line called the guadrat, & which hath two ſides, divided each into % 100 equal parts; and that on the left hand is called right ſshadow, and that on the right hand contrary ſshadow. By „ which means tis very eaſy to take any 4 acceſſible altitude at one ſtation by the ce quadrant. For, if the angle be juſt 455,

the diſtance to the foot of the chje& from

the ſtation is equal to the altitude; but if the angle be leſs than that, the ſtring will fall on right ſshadow in the % quadrat. Then ſay, as 1oo is to the

number of parts of the right ſshadow, cut by the ſtring, ſo is the diſtance to the alti-

tude. 80 that if the ſtring cut 25, 75,

Lexicon Technicum, Vol. I. Title Altitudes.

or

r 272832 2-322..7

tr or 50, &c. the height is. ; accordingly : 1, 25 t or 2 the diſtance. But when the ſtring falls towards the right hand, and the angle be above 4.5, then ſay, as the parts cut by te the thread in tHe quadrat are to 100, fo is the diſtance to the height. Where= fore, if the thread cut 50 here, the height jg double to the diſtatice; if it cut 2 55 & it is four times as great as the di- ſtance, Gc.

We readily agree with the leamud Doctor in this reſpect, provided the object be perpendicular, and the ground horizontal: for altitudes of objects are always meaſured by the neareſt diſtances from their horizontab or level baſes. But, on the other hand, if either the object vary from a perpendicular poſition, or be ſituated om ſwanting ground, there muſt be an error; my be RO thus:

2 — 3 5 : 1 ; I " F 1 , 8 A , #.\,.5 04 Xample 224 I & 7 ; - " 4 -

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8 x" +4 1 — 1 renn I

Let Cx repreſent a tree ſtanding upon the horizontal baſe An B, whoſe reclination n from the ee being our to find the length A C.

Make an obſervation at A, cauſing the plummet to hang directly over the 45th degree; then ſhould Ar be equal to Cx by

the firſt general rule: But CM An+nv. +Cz—Cv; as is evident Rk the above 3 figure. ILLUSTRATION. Suppoſe Az=20F. and vv 4F. then is An+nv= 24F. Now, the angle at v as

being a right one, and the angle of obſervation at A being 45, the ACy, CAv

—

CAy will be equal, Corol. 4. Eu. 32. 1. Equal angles alſo ſubtend equal fides, Eu. 6. 1. Therefore, Auf Cv 24 F.

Again, JT" + 24F. 31. , Ge. Eu. 47. 1. B the following rule 5 finding the length of the tree: To the ſquare of the altitude Cv, add the ſquare of the reclination 7 v; the ſquare

root of that ſum will be the length of the tree Cn 1 5 5

e n IL N Other lag remaining as before, let B now be the place of obſervation, (the plummet ſtill cutting the 45th degree) and the horizontal line B v being equal to 24 F. 10 find the length of the tree C M.. Here BA ſhould be equal to C by the firſt general rule: But C BAA vT C —Cv; as by the above figure.

ILLUSTRATION. Now Bu being equal to 28 F. and v 4 F. then By—zv=24F. The angle at v 18 a right one, and the angle of obſer-

F$J vation at B is 45*; therefore the angles 'v CB, CBy are equal, Corel. 4. Eu- 32.1, Equal angles ſubtend equal fides, Eu. 6. 1, Hence BA — 2 v SC F. Again,

y . 24 F. 31 35 Es. Eu. 47. : From this illuſtration it follows, 2 the ſquate root of the ſum of the ſsquares of the altitude Cy, and reclination vn, gives the length of the tree Cn the ſame. as before. Having ſhewn the inutility of the us grant when applied to meaſuring trees, in en oblique poſition, ſtanding on horizontal baſes; we ſhall next proceed to point out ſuch errors as will ariſe in the computation, when they are ſituated on OY or Frading planes, 1053

Example

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B 8 .S.

Let Cp . a tree landing perpendicular to the horizontal line Ax, upon a 4 deſcending plane Ap B, whoſe depreſſion p # being given; to find the length of the faid + tree. Make an obſervation at A, cauſing the plummet to hang directly over the 45th degree: then ſhould A þ be equal to ꝓ C by the firſt general Rule: but Cp=&p + 2 þ An by the figure,

Illustration,

Suppoſe Ap equal to 14 F. and the d.

preſſion 5 equal to 2 F. then IF

e BALED Eu. 47. 1. The angle : 34 at

Ase

at n being a right one, and the angle AC 45*, it follows, that AC is alſo 45*, by Corol. 4. Eu. 32. 1. And as equal angles ſubtend equal fides, Eu. 6. 1. therefore An n, and An +np= Cp= 15 F. 11 1. Hence the following rule: 5 From the ſquar, re of the oblique haſe Ap, ſsubtract the ſquare of the depreſſion Pn,

and extract the ſquare root of the difference;

Et

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the root thus found, being added to the given ; depreſſion, 8 the N 1 of che 5 tree.

t

E x A M PL E IV. 21 c þ Bag" ting the tree, as before, let t the obſervation be made at B, the plum- - met , ſtill cutting the 4 5th degree of the quadrant, then ſhould By be equal to C þ, by the firſt general rule: but Cp Bp

nr by the figure.

„ Let By be equal to 19 F. 5 I. and pm (the elevation of the aſcending plane Bp A above the horizontal line Bn) equal to 3 F.

31. w NF — ad- neuer. 21. ' Eu.

Eu. 4. 1. Now the angle Cn B being 90, and the angle mBC 45e, the angles

„BC, BC n are equal, Corol. 4. Eu. 32. 1. But equal angles ſubtend equal ſides; there-

fore C B 10 F. 21. Eu. 6. 1. Hence Bm—pm=Cp=15P. 111. From this

illuſtration the following rule is derived:

From the ſquare of the oblique baſe Bp

ſsubtract the ſquare of the elevation #2 Pb.

and extract the ſquare root of the difference;

from this root ſsubtract the elevation, and

the remainder is the length of the tree. Thus having ſhewn the inſefficiency of

the quadrant, in meaſuring the length of trees not exactly perpendicular to the horizon, nor ſituated upon a level baſe; it ſeems

altogether needleſs to add, that neither the theodolite, nor any other graduated inſtrument of the like nature, can give the true meaſurement of objects, ſuch as theſe,

whoſe forms and ſituation are 2 fo extremely various and uncertain.

The author of this work Why had

frequent occaſion to obſerve the uncertainty of meaſuring ſtanding timber, unaſſiſted

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b any proper inſtrument for that purpoſe, conceived his preſent deſign of rendering the practice more certain and familiar, both to the owner and ſurveyor. In this purſuit many difficulties have occurred: how far he has obviated them, is humbly ſsubmitted to the reader's determination.

The inſtrument now offered to the public, called a DENDROMETER,— for which His Majeſty has been pleaſed to grant His Royal Letters Patent, —is an invention entirely new. It is founded upon that branch of the mathematics called plane trigonometry, and its principles are chiefly derived from the 2d, 4th, 5th, 6th, and 33d propoſi- tions of the fifth book of Euclid. It is formed to meaſure, with great accuracy, not only the length and diameter of any tree, in any situation, but alſo the length. and diameter of the boughs, by inſpection only; and thence by the help of tables

(hereunto annexed) the certain quantity of

timber is obtained without calculation or the uſe of the sliding-rule. By this means a criterion is fixed between the buyer and

ſeller of an article, which becomes, every day, more and more valuable; as either party will have it eaſily in his power to prevent error, or to detect impoſition.

'Bat, notwithſtanding the meaſuring of ſtanding trees with eaſe and certainty may be eſteemed no inconſiderable acquiſi- tion; yet, we preſume, that this inſtrument, when fitted to a theodolite, will be found applicable to more extenſive and important purpoſes : vix.

For the practical meaſurement of the heights and diſtances of objects, acceſſible or inacceſſible, whether ſituated in planes parallel, or oblique, to. the plane in which the inſtrument is placed.

For the taking of all angles, whether witical horizontal, or oblique, in any poſition of the planes in which they are formed; whereby the fides and angles of all plane triangles, right-angled or oblique, are determined upon the inſtrument, without any trigonometrical computations, from the uſual data; and that, whether level to the line of ſtation, elevated above or

— - — below the ſame, acceſſible or

inacceſſible, upon their own planes, and

alſo upon the plane of the horizon.

From the above circumſtances, it need not be urged how uſeful it will be found in field- operations, particularly in the practice of engineering and land- ſurveying: For, by the method of applying the altimeter, elevation- index, and other moveable parts of the inſtrument, the meaſure of the required ſides and angles (right or oblique)

1s. determined to ſsufficient exactneſs 201th

out, any computation; or the uſe of tables, hitherto neceſſary to aſſiſt the theodolite and all other graduated inſtruments.

In the practice of engineering, we apprehend, it will be found particularly uſeful;

as not only the diſtance from any fortification, but alſo from any viſible part or projection thereof, together with the ſituation of the plane and place (with reſpect to the fortification) whereon a battery 1s proper to be . raiſed, may all be readily obtained, without being under the neceflity (according. to the preſent mode of practice) of advancing

Pw

+dancing upon the ſpot, or of approaching

too near to the fire of the enemy We alſo conceive, it will be 3 exceedingly uſeful to a field officer under the following circumstance. Suppoſe a body of the enemy are perceived, by the

General, advancing on full ſpeed towards

his army, and no time allowed for making variety of obſervations : by means of this inſtrument he can quickly diſcover (by lines drawn from the points of obſerva-

tion to the extremities of the rear of

that body) the angles made, and thence the number of ſsoldiers contained therein, which will enable him to direct, with very little OY what force to march againſt them.

Then its uſefulneſs in 2 eſpecially on hilly or irregular ground, we apprehend, will be evident, when our readers are informed that the perpendicular riſe or

fall of the land, the hypothenuſe, and baſe-

Iine, neceſſary for protraction, are determined at once upon the inſtrument with- 55 cu out

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[14] out calculation, and coſegnently the riſk of error. ma $740 828 E

In the courſe of this FRY . oe were omitted, and others which appeared not ſsufficiently clear to our judicious friends, are added by way of Ap PREN- Dix. But ſtill, if there ſhould remain any obſcure . paſſages, we ſhall with pleaſure endeavour to elucidate them, at the revueſt of any perſon, who will give himſelf the trouble of pointing them out.

We ſhall now proceed to give a particular deſcription of the inſtrument, with Plain directions for the uſe of it. The Tables annexed are calculated to ſhew the whole quantity of timber contained in any tree, of any diameter, girth, and length, both by the exact and common method; by which means, the error in the uſual way of meaſuring round timber, ſo much

to

They are carried from 6 in. to 80 f. in length, and from 6 in. to 4f. 3 in. + in diameter, and are fo con-

trived as to admit of being eaſily extended to any further dimenſions,

to the loſs of the ſeller is diſcovered and corrected. To conclude, if, by this invention, the meaſurement of ſtanding trees be rendered eaſy and certain, and the practical operations of engineering, ſurveying, levelling, mining, Cc. greatly facilitated, we flatter ourſelves it will meet with encouragement ſuitable to its merit and importance,

It is obvious, that the preſent method of meaſuring timber is greatly to the diſadvantage of the ſeller : for, by taking the girth in the middle, and + of the circumference in inches for the ſquare, the larger the tree, the greater muſt be the loſs of the ſeller, even to of

the contents in a tree of 48 inches circumference.

Description

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CLXEELENSEXELEALEXENE LY 1 XS ts, r ate OH Ya Ka HAK SR Ae ELTEN

Description

Oy THE

Dendrometer.

OLE HIS inſtrument (which gives

P 2 direct and general ſolution of the whole 7thirteen caſes in Xx plane trigonometry, without calculation) 18 cotnpoſed of divers members or Parts, VIZ, 1. A ſemi-circle (A), which is divided into two equal parts or quadrants. The circumference of theſe quadrants are each divided into ninety degrees, &c. beginning at the middle of the ſemi-circle ; and upon the diameter (B) thereof hangs a plummet

Ws

3 K 12 ] (C), in order to fix the inſtrument in a vertical poſition ; which may otherwiſe be effected by means of a ſpirit-level. There is alſo a chord (D) parallel to the diameter, and likewiſe a radius (E), which paſles at right angles through the gliameter and chord.

2. From a point on the, radius hangs an altimeter (C), between the chord and diameter, to which is fixed a ſmall ſemi-circle (GY, and a ſcrew to confine it in any poſi-

tion. The altimeter, which is contrived

to form the ſame angle with the radius of

the inſtrument as the tree forms with the

borizon, iS divided from. its center both

ways, each into forty equal parts; and theſe

again, s by {ſmaller diyiſions, into halves and quarters. Upon the ſma!! ſemi circle (G), an which is accounted the quantity of the angle made by the altimeter and radius, are expreſſed from 60 to 120 degrees, being 30 on each quadrant. The radius is numbered with the fame ſcale of diviſions as the altimeter. There is allo a nonius to the ſmall

De Oa) Tt

E uz

em- circle, which ſhews the won of an

angle to every five minutes. 3z. On the back of the TER 4 ock (W) of the ſliving- piece is confined: to the axis (N), which moves concentrically parallel to the elevation-index (F) on the oppoſite ſide, it being confined thereto. This index is numbered by a ſcale of equal diviſions with the altimeter and radius. At the end thereof is a ronzus, by which the angles of elevation : above, or depreſſion below, the horizon (meaſured upon the ſemi- circle of the inſtrument) are determined to every five minutes. There is alſo a groove in the radius, that ſlides acroſs the axis by means of a ſcrew (1), working between the chord and ſemi- circle of the i in- ſtrument. 2 e 4. Upon the ſtock i is a ae var (S* that always acts at right angles with the altimeter, by means of a groove in the latter. To the ſhank of the ſliving- piece is affixed a moveable limb (Q), which forms the ſame angle with the altimeter, as ö the

e

the bough forms with the. body, or trunk? of the tree. This limb may be of any convenient length, divided into equal parts

of the ſame ſcale with all the foregoing

diviſions. At the extremity of the fixed axis, on a center, an index (R) with tele- ſcopic fights works horizontally upon the moveable limb of the sliding-piece.

Upon this horizontal index (R) may be fixed a ſmall quadrant (T), (deſcribed with any convenient radius from the center whereon the index moves) divided into go degrees, beginning at a right line drawn from the center at right angles with the fiducial edge of the ſaid index; and upon the extremity of the axis a nonius, whereby to determine the quantity of an angle upon the quadrant to every five minutes. There are alſo two ſnall circular arches (S), - ſerving to keep the fights in a parallel poſition, each containing an equal number of degrees. Upon theſe arches is meaſured the angle, ſubtending a ſide 8 n to the + e of the altitudes of the

obſerved

ra

d

obſerved objects above the plane of the horizon, and whoſe baſe is the neareſt diſtance between the perpendiculars, in which the ſaid objects are ſituated.

N. B. The Dendrometer is fitted to a theodolite, and may be uſed either ſseparately therefrom or jointly therewith, as choice may direct, or occaſion require.

Ls

Reference

215 J

N 91S 70 the Plant, ;

A B D E E. G H

The demi: circle. The diameter. The altimeter. The chord. The radius. The elevation- index. The ſmall ſemi-circle on the altimeter. Stays to the altimeter. The ſcrew, that draws the radius towards and from The ſtay, by which it is confined to the ſpring-ſocket or theodolite. The plummet. N The ſtock of the ſliving- piece. The axis. The key to move the ſerew of the radius. P The ſliving- piece. „ 4 Q The moveable limb of the Lg pes R The horizontal. index, on which the teleſcope i is fixed. S Small circular arches to 3 the moveable limb of the flidin 85 piece, and the horizontal inder, in any poſition, Sc. T The ſmall quadrant the horizontal

index.

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& 8 - ) : — Sk . 6 / K+

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The 1 Form Fo a. 1 -Book

Vor Standing Timber.

k 4

17 E

Height.

—

KEE ah 41 — 3 — —

0 "

ey " ö ey + a " '

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— — — — a %

—

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— 14 —

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— — — — 2 — 1 — —— — 4 „

-

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1 2 Sts Isilles +4 +2

N

Diregtions

1 ver .

INSTRUMENT. —— Ln

CH A 2 To e, 4 tree in any e on level ground.

ps the Ut at any

7 convenient diſtance from the

N tree; then, by means of the He plummet, ſet it in a perpendicular poſition ; which done,

1 it faſt upon the ſtaff. Set the altimeter in the exact poſitian of the tree and ſcrew it faſt. With your tape-line

That is if the tree recline or incline, ſo much - muſt the altmeter recline or incline; ot, if the tree be perpendicular the altimeter myſt alſo fe perpendi-

cular. meaſure

tree to the ai of the inſtrument, and with your key move the radius, until that dis ſtance be cut upon it by the inſide of the diameter. Thus is your inſtrument properly

rectified

7 o take the tneeh of a tree. Put your left hand upon the end of the elevation index; and your right upon the ſhank of the ſliving- piece cloſe to the

ck, on the under ſide; then move the in-

dex downwards, until you ſee, through the ſights, the bottom of the tree cut by the horizontal wires; and there note the feet and inches marked by the index upon the

altimeter below the point of fight or ho-

rizontal line. Move the index upwards, until you ſee the part to which you would

meaſure, cut likewiſe by the horizontal wires; and there note alſo the feet and

inches marked by the index upon the alli-

In the form of a watch-key for that purpoſe. + The inſtrument is to be rectified thus in all mod As far as the timber extends.

C 2 meter,

meaſure the horizontal diſtance from the

length of the tree; which inſert in the {e- cond column of your field-book, and then

through the fights, the left fide of the tree

of the body of the tree; and as the diſtance of that

Fey: meter, above the point of fight : theſe two quantities added together, give the exad

proceed 0 i 04 To take the girty.

With your tape: line meaſure the circumference of the tree in that part whence you took your horizontal diſtance; and let the diſtance on the radius, which was before cut by the inſide of the diameter, be now increaſed by one ſixth of the above- mentioned circumference this done, lower the elevation-index to the part of the tree, where you purpoſe to take the diameter; and ſcrew it faſt. Set the moveable limb of the ſliving- piece quite ſtraight, and the edge of the Horizontal index upon the firſt division thereof. Turn the whole instrument about to the left hand, until you - ſee,

1 As the tape-line cannot be applied to the center

point to the circumference is nearly one ſixth part thereof, the above allowance becomes neceſlary.

cut

cut exactly by the Perpendicular wires: then (the inflrument being fixed) move the ſights only upon the sliding-piece, until you ſee the right ſide of the tree cut alſo by the perpendicular wires; and you will find the true diameter marked by the horizontal index upon the Aiding-piece. Having entered: this in the third column of er fieldbook, you may proceed 0

—

Zo meaſure the FS

As the radius was increaſed by a fifth, part of the circumference to take the diameter, it muſt now be reduced to its former quantity, viz. the diſtance from the tree to the axis of the inſtrument. This done, move the elevationindex upwards, until the bough becomes viſible through the fights, and there ſcrew it faſt. Set the moveable part of the //{dingpiece in a poſition parallel to the bough, with the edge of the horizontal index placed upon the firſt diviſion thereof. Turn, the whole inſtrument about, until you ſe, thro on the shoot of the bough, cloſe o the C 3 zunk

22 J trunk of the tree, cut by the perpendicular wires; then move the fights, until you fee the other end of the hough cut by the ſaid wires, and there note the feet and inches marked by the horizontal index upon the moveable limb of the /iding-piece ; which

will exhibit the true length of the bough. Having entered this in your field- book, you may take the girth of the e in the fol lowing manner.

Direct the ſights to that part of the bough in which you intend to take the girth; then moye the elevation- index downwards, until you ſee the under ſide of the bough cut by the horizontal wires; and there note the feet and inches marked by the ſaid index upon the altimeter ; this done, move the elevation-index upwards, until you ſee the upper fide of the bough cut by the herigontal wires; and there allo note the feet and inches marked upon the altimeter 18 before: ſsubtract the former quantity from the latter, and the remainder will be the true diameter of the bough ; which enter in bel book. In like man- ACE - ner

N tas LEONE: bo N N — ß ß £ g : OO Sa or op RS T_T, : A

J (ä

. $ 8 3 Kor "Au Tu EY 3 ;

Es 1

ner the reſt of the boughs are to be mea- ſured, obſervin g to place the inſtrument where they appear moſt open to the

fight, and properly adjuſting it, as oc-

caſion and change of ſituation may re-

quire. Having thus ſmiſhed the . whole proceſs of meaſuring, turn to your tables of timber-meaſure ; look at the top for the

diameter or girth of the tree, and under it, in the firſt column, for the length; and oppoſite thereto, in the ſecond column, is the

true ſolidity or real quantity of timber contained, in the body of the tree; and in the third column you will find the common ſo- lidity, as given by the ſliving- rule: and fo likewiſe of the boughs; all which may be

transferred into your field-book at leiſure.

4 Example

r

4 2G Q3- 9538-2512

WV, Ys E 3 4 8 1 12 . n — o 0 2 432881 4 IP ; 4 %

7 . a s ©

— — —

1

Let EBC repreſent a tree ſtanding upon the level ground G D. Place your inſtrument in a perpendicular poſition by the plummes at any convenient diſtance, as at A. Set the altimeter in the exact poſition of the tree. With your tape- line meaſure the diſtance from the tree to the center of the inſtrument, viz. BA (which we will ſuppoſe to be. 17 feet). Move the radius

%-

either

„ &

&

. ä An wad FREY 5 9 1 ä gets £ x G50 2 Ac g 3% Sl I „ 5 A 2 0 7

Kod

W

ber cut by it upon the ! altimeter down wards from the center (ſuppoſe five feet);

25 feet: add theſe two numbers together,

place B whence you took your diſtance, and . increaſe the number on the radius by one ſixth of that circumference: then depreſs the elevation - index to that part of the tree where you deſign to take the diameter, as

1 25 ]

either backwards or forwards, as there

may be occaſion, until that number of feet

e . , the . ben direc ha Bebe to ebe keenly

this: elevation-index,.and there note the num

this done, elevate the ſights to the direction AE, and there alſo note the number cut upon the altimeter by the elevation- index, from the center upwards, which we here ſuppoſe

and their ſum (30 feet) will be the whole length of the tree; which enter in the ſecond column of your field-book, -

Next, with your Zape-/ine . che circumference or girth of the tree, in the

. pe I 4 — — Y - n p ks uf 2 A . 1 — ® i 8 pj 8 — — — — bo —— Cf P ö 5 Q py - —

. —— 2 n

at no, and ſcrew it faſt: lay the moveable mb of the hiding piece quite ſtraight, and the

Horixontal

46, D

k

q

126 ] herizontol index upon the firſt diviſion there until, through the fights, you ſee: that ſide of the tree cut by the perpendicular wires :

then move the fights; until you ſee the other fide of the tree cut in like manner by 8. the wires; note the number cut by the Bo- Ft the Hiding piece, and you will find the true diameter of tha tree x o, marked thereon, (ſuppoſe four feet,) N the third column of your field-book The girth being thus 8 the radius back to the former diſtance, wiz. 17 feet: then raiſe the elevationindem to the direction of the bough EF, which you deſign to meaſure, and there ſcrew it faſt: ſet the moveable limb of the hiding-piece in a parallel poſition to the bough, and lay the horizontal index upon the firſt division thereof. Now turn the whole inſtrument about, until, through the ſights, you obſerve the ſhoot of the bough at E, cut by the perpendicular wires: then move ng art Ka tremity

te?

cremity of the bough, at F, cut alſo-by the perpendicular wires; and there note the number marked by the horizontal index upon the moveable limb of the sliding-prece, which determines the true length of the bough: this done, direct the fights to that part, as 8 v, of the bough where you intend to meaſure the diameter; depreſs the elæuatian - index, until, through the ſights, you obſerve that point v in the under ſide of the bough which is cut by the horizontal wires; and there note the number marked upon the altimeter by the elevation-index ; then raiſe the ſaid index, until, through the fights, you ſee the upper ſide of the bough, at 8, cut by the horizontal wires; there alſo note the number marked upon the altimeter: ſultra& the former number from the latter, and the remainder will be the diameter of the bough : then refer to your tables, as before-mentioned, for the ſolid content, With regard to the uſe of this inſtrument, it being neceſſary in all cafes to adjuſt it according to the directions at the be-

þ ginning of this chapter, and the whole fual ture

. as may ſerve to obviate whatever: difficulties are likely to occur, in finding the length of trees ſituated on ſwanting

bs: , ra pre trees, Se. and likewiſe

A the interven-

: non of à river, . l bſtacle. TY

E "

o 1

9 d — I Fa F.þ 2 — —

— ys — 4

+S f N . - — — LY 1 a hs Pa 7 5 9 o — - s on

2 we ſhall proceed to give ſuch

Rn a .

22 As +

—

R

ee e

1

To meaſure @ tree in any poſition on Hunting ground.

— a wil „ p SB IE TT CO LOTS 1 FW

— . 3 1 © " th,

tk dp e r 8 F 7 n wy

Example Ii.

F

Lies

£ P NE n 8 1 — 5 5 ESO bf ef.

ET OXS repreſent a tree ſtanding upon the planting baſe NM: place

your inſtrument at any convenient diſtance, as at L, and there fix it perpendicular to the horizon ;

A. ; of

6

N

'F

? þ %

A

q

—

L +, Diet

„„ Fg

horizon; ſet the altimeter in the exact poſi- tion of the tree, and ſcrew it faſt. This done, inc aſure the diſtance, X L, from the tree to the inſtrument with your rape- line,

which we here ſuppoſe 24 feet: move the

elevation- index, until, through the ſights, you ſee the point of the tree from whence you meaſured, and there ſcrew it faſt: then with your 4#y move the radius either backwards or forwards, as may be neceſſary, un-

til the aforeſaid diſtance be cut upon the

elevation-index by the perpendicular line of the altimeter ; and the horizontal line L 75

cut upon the radius by the inſide of the

diameter, will be found to be 19 feet.

The inſtrument. thus rectified, the re-

maining part of the work muſt be performed according to the rules laid down in the firſt example.

Note, If your lation be lower than the

root of the tree, and the numbers cut by the elevation-index upon the altimeter be, at the ſame time, above the horizon their

difference b

By the horizon is meant that imaginary line, which: paſſes through the fights when the inſtrument ſtands

difference gives the true length of the tree. On the contrary, if your ſtation be higher than the top of the tree, and the numbers cut by the ele vation- index upon the altimeter be below. the horizon, the length of the tree will be equal to the difference of thoſe numbers, as before.

Moreover, if one number be found above, and the other" below the horizontal line, drawn from the point of ſtation to the tree;

in this caſe their ſum exhibits the true length of the tree. IFC

If your ſituation be ſuch, that only one number be cut by the elevation-index upon the altimeter, (and it falls either above or below the horizon) hat number expreſſes the exact length of the tree.

ſtands perpendicular, and the elevation-index points to zo degrees: or, in other words, when it cuts the center diviſion on the akinuter, and lies . to the real horizon,

Chap.

f af

-

——

a 2 n - 2 70 8 8 Tc

—

+ — ee one te Ne. ++

Akn.

Chap. Iii.

To meaſure an irregular tree, fituated upon .., offending or deſcending ground.

$533

£2125 2% nn IM.

#' St a4 # + 4+ 9425 ++

. 9 — 4 - 5 : &

inſtrument perpendicular to the horizon at a convenient diſtance FC, or GC, and ſet the altimeter in the exact poſition of the lower part (B C) of the tree; then with your ta pe-line meaſure the diſtance u F,

or

ET ABC repreſent a tree ſtanding i upon the ſwanting baſe DE : fix your

?

of 1G, and proceed according to the rules laid down in the ſecond. chapter, IP, This done, and the meaſurement. of the lower part of the tree obtained, raiſe the elevation-index, until you ſee the point B*; ſcrew it faſt, and, obſtryr the num ber cut upon the ſaid ind by the per- Pendicular line of "the "altimerer, Set the altimeter in the exact Holition of che upper

"I

part (AB) af . the... tree;.. crew. it faſt, Move the raus backwards at: forwards, 48 occaſion may require, until the number defore obſerved on the Jevation-index," 'be

Wh TY

again cut by the perpendicular, ling. of the altimeter. Thus is your inſtrument. reQtified for the ſecond operation, muſt be performed as follows. . Obſerve the. number cut upon, the alti- 5 meter by the eiruation- ind, when the

ſights ate directed to B; and alſo the num ber cut upon che altimeter by the ſaid ndex, when. they are directed to A: the former of theſe numbers ſubtracted from

e height to which you have already

D the

1 „

che latter, gives che length, A B, of the up⸗ per part of the tree. The diameter, and

likewiſe the boughs, muſt be meaſured 45 befote dir rected 1 in the firſt example.

5001 230 1 14

F 7 1 þ 7 " 'C H A p. 4 axgh® ce inn a 112 AL + 44

14. 99014 - 0 a + Sd

lt maſt here be 1 that there will be a ro error in this meàſurement, from the circum-

ſtance of che crooked part at B: for, if you meaſure on the exterior fide of the. crook (from F), then is the length, already taken, more than the mean length, by balf the tangent of the angle of the tree's reclination from a right line. On the other hand, if you meaſure on the, interior: fide of the crook (from G), then the length, already taken, will be leſs than the mean "length, by half the tangent aforeſaid. Now this =error, thöugh very 'trifling, may be corrected, and -the true meaſurement taken i in the 1 manner;

vin. id n c — T. the length of the tree, as before di-

3 both on the exterior and interior ſides of the - crook;; add therm together, divide the ſum by two; and the quotient will be the mean length required. Or, 2. When the altimeter is ſet; in the poſition of the ſecond part of the tree (A B), obſerve the angle of the ttee's reclination or inclination upon the ſmall ſemi- "tirdle of the altimeter, and alſo meaſure the diameter of the tree in the crooked part. T his done, ſet the altimeter in a perpendicular poſition, and the elevationindex to the angle of the tree's reclination or inclina-

tion,

[ 35 1

Hap. Iv.

To meaſure a tree, inacceſſible by means of 4 river or obſtacle intervening. .

AVIN G treated particularly upon

the meaſurement of acceſſible altitudes, it may not be unentertaining to our readers, to ew. in. the next example, the method of meaſurine 81 ſuch as are inacceſſible.

tion, obſerved before on the ſmall n of the altimtter : move the ſcrew, until the diameter of the tree in the crooked part be cut by the diameter.-of the inſtrument upon the radius: and you will find. the tangent of the angle of the tree's reclination

or inclination cut by the elevation-index upon the alti- © meter, Add half this tangent to the length of the tree, if on the interior {ide of the crook, and the ſum

will be the mean length required. On the contrary, if on the exterior ſide of the crook, ſultra& half this tangent from the length of the tree, and the difference will be the mean length required.

D 2 Example

—— 7 - — 3 — 1 Ws 1

—

—

——

—

2 of

I — Et

a — , n

——

— — „

Place your inſtrument. at any convenient

il 36] HE Ax x; I

Let A reprefent a tree, rendered inacceffible by the intervention of a river.

Point, as at, D, and take the angle BDA. enter it in yaur field- bock. Advance to O, and there take the angle BCA; which alſo inſert in your field- ook: meaſure the diſtance CD; and inſert it in your book: ſsubtract the angle of altitude observed at D, from the angle at C, and the remainder will be the angle CAD: then the

will be, I. As

17 ai kw ac

N I 1. As the ſine of the angle at A is to the diſtance CD between your ſtations; ſo is

the ſine of the angle at D to the length of the line A C.

2. As radius is to the length of AC ; i is the fine of the angle at to the height A B of the tree.

N. B, We are aware that this method i is only true when the obje i is perpendicular to the horizontal or level ground, between the ſtations. But it mult be obſerved, that 7 this is the uſual method practiſed in engineering, to obtain the height of any fortification. To obviate every difficulty, when the ground between the ſtations 1s irregular, the inventor of this has conſtructed another inſtrument for the uſe of His Majelty's engineers, which will readily determine, without calculation, the true height and diſtance of any tower or battery, or of any part thereof; and alſo its projection or depreſſion, either above or below the hotizon.

D 3 „

cH AP. V. 15

Hane already mentioned the dif. ficulties attending the ſurveying of hilly, or uneven ground, with the 7heodolite; and remarked that, amidſt ſuch a variety of calculations, ſsurveyors may be liable to dangerous errors; we ſhall further beg leave to obſerve, that notwithſtanding Mr. Hun & (editor of Wilſon's Surveying) ſeems to reject the uſe of the 7heodolite, and other graduated inſtruments ; yet we can by no means agree with that gentleman, in recommending the uſe of the chain alone, as fully ſsufficient in a// caſes. On the contrary, we poſitively affirm, that no eſtate or lordſhip can be correctly ſsurveyed by the cham alone, unleſs the land be truly level. For in mea- ſuring over aſcents and deſcents, it is evident, the meaſurement of ſuch line will be much longer than the horizontal or baſe line; and,

In the Introduction. § In his Argument, p. 291.

conſequently,

£6

Ft.

conſequently, no true plan can be formed, without the aſſiſtance of ſome graduated i in- ſtrument, to take the angles of acclivity or declivity, previous to calculation. 53 Rai

We ſhall now proceed to illuſtrate by compariſon, in the following chapter, the advantage of this our inſtrument oyer the theodolite, &c. in the practice of surveying,

particularly of hilly ground,

'Chap. Vl

The proceſs of ſurveying billy ground with the theodolite, &c. from WiLsoN's Surveying, Sect. I. p. 1

Ky - ( : 8 F , o So 1 TS F iy "T's 4 £0 Py %

'E'T the mountain BAD, be ſup-

' poſed to ſtand upon the horizontal

„line, or plane BCD, and its perpendicu-

lar altitude AC, and the whole horizon 6 'M 2 of the plane BD is required.

De " ring

BR. Bring the inſtrument to the bottom of " the mountain or hill at B, and let your aſſiſtant go to the top of the hill at A, and place a mark that may be ſeen from B, and elevate your. inſtrument, ſo that oe you can, through the ſights, ſee the mark at A, and obſerve. the degree of altitude, 6 which ſuppoſe 37 deg. go min. *The angle ABC, found as above, being „ 37 deg. 39 min. the angle BAC is 52 deg. 30 min. (by Theor, V. the angle BCA being a right-angle:) therefore to find the baſe B C, by caſe the foe of right e "angled plane triangles, OG

cc As radius 10. o

To hypothenuſe BA, 5 „ chains, or 330 feet, 2.51851 ce 80 Sine of. BAC—52; 30 9. 92945

60 . To BC, 262 feet 2. 41797 as Then proceed to the other fide of the

% mountain or hill at D, and with your in-

ſtrument take the angle C D A, as before,

which ſuppoſe 56. deg. oo min. and let the „ fide; NA, when meaſured, be 270 feet, 6e ee find the baſe CP, as before. N

—

The .angle D being 56: oo, the angle DAC will be 34: 00 to find CD.

% As radius 2 0 Io. ooo

wh „ ee — 2443136

1 298 4 b 6 # + 1 4 Wa 4 4

to SDS feet —— . 0 The Line BC 26 feet.

Adder to CD rasen 151 feet.

70 Thefami 18 the whole line B D 41 3 feet,

VN. B. Mr. WI leon has omitted the celeulations for the giſt N

The be proceſs of — the fame bill bees

worth the een och: ee

Let ABD e ame: mountain

25 before, whoſe baſe and perpendicular are

the lines BD and AC reſpectively.

Place your inſtrument in a perpendicular

poſition, at B, and at A ſet up a mark, of

equal height with your inſtrument. Raiſe the elevation-index, until you ſee the mark

at A through the , and there ſcrew it faſt; 8

9

faſt: then with your chain meaſure B A,

the fide of the hill, {which admit to be 20 poles, or 330 feet, agreeable to Mr. WII-

sox's example.) This done, ſet the altimeter perpendicular to the horizon, and ſcrew

it faſt. Move the radius, either backwards or forwards, as occasion may require,

until the number above- mentioned be cut upon the elevation-index by the perpendicular line of the altimeter; and vou will find the ſsegment BC, of the baſe⸗ line BD,

to be 1 5 poles and 22 links, or 262 feet, cut

by the infide of the diameter upon the radius; likewiſe the perpendicular height AC to be 12 poles 4 links, or 200 feet 7 inches, cut by the ele vation- index upon the altimeter. Now remove your - inſtrument to the top of the hill at A, and direct the elevation-index downwards, until, thro! the ſights; you ſee the mark at D; having ſcrewed it faſt, meaſure AD with your chain (which ſuppoſe to be 16 poles 4 links,

or 270 feet). Then proceed as on the other

fide of the hill, and you will find the ſsegment n of the baſe BD, to be 9 poles 4 links,

t 43 1

4 links, or 151 feet; The ſum of theſe ſsegments of the baſe gives 25 poles 1 link, or 413 feet, for BD, the baſe-line required.

From hence it is manifeſt, that a ſeries of abſtruſe calculations is avoided, and, by that means, the foundation of error removed; as the true meaſure of the baſe, the perpendicular and bypothenuſe, are all marked at onge upon the inſtrument; whereas the calculations, requiſite to aſſiſt the theodolite, &c.-(to ſay nothing of the time unayoidably employed in the operation) are liable to many errors.

Thus having, we apprehend, clearly demonſtrated, hy this comparative view, the advantage of our inſtrument over the theodolite, &c. both with regard to eaſe and expedition ; ; we ſhall farther illuſtrate its uſe

in meaſuring hills ſituated upon inclined haſes.

Chap.

i — — r O22. ©

N .

N Vii.

Concerning the meaſurement of a 8 or. Bill, when the ground on each fide thereof #5 not in the ſame horizontal plane.

E T AB 1 DE eee the plane ſurface on the oppoſite ſides of the hill BCD: then will BD form the oblique, and Dy the true horizontal baſe. Now take an obſervation at B, Cn will 'be the perpendicular height, and By the ſsegment of the horizontal baſe. On the contrary, Cg will be the perpendicular height, and Dꝗã the other ſsegment of the baſe, if the obſervation be made at D. We may here obſerve, that if a level from the plane ſur-

face

ä face A B was continued through the hill, ſuch level would appear above the opposite ſurface DE, by the perpendicular height

ey. On the other hand, a level carried from the ſurface of the plane DE, through

the hill, would fall elo the oppoſite ſurface AB, by the perpendicular depth Bg. From hence it is evident, that BD does not form the horizontal baſe, nor can Cx, or Cg, be the true height of the hill. But the distance from the ſummit C, to the interſection of the lines BD and Cc 7. determines the height of the hill. Note, As the horizontal baſe D g, is to the difference Bg of the heights 7 C, nC, fois the ſsegment D'g to a fourth number,

which taken from the perpendicular 4 7. leaves the CEN required. '

Chap.

kit F $1 DS "CHA pP. VIII. 7

SH af the gefl. of an irregular . DNS CC. g "5 1 — + 7 2 19 I 4 © % : + £ 'CL 4 6 4 9 4 ; , - 2 . E — 1 IS F +; 143 3 41 il f L . . . 0 i b . Fo / ET ABCDEF be che boundary of b :

an irregular field; the dotted lines . the riſe and fall of the ground. Firſt, cauſe large and conſpicuous marks to be erected in all the angles of the field, at an equal diſtance from the hedge, making your obſervations without the field. This done, remove the mark from A. Fix your inſtrument in a perpendicular position, in the ſame place where the mark ſtood. Set the index of the 7heodolitepart upon the 360th degree. Turn the whole about, until, through the ſights,

you

5 3 3 7 1 F . N . & — EY N 2 1 1 n .

' „ ys . — A8 6 1 „ . . 0 r

5 N A n 5 n Y E TED I LS Gs - 338 You they 8 68 AS 8 WC II . ̃ SE IBER ESR ß 2 a

part, until, through the fights, you ſee the

[ 47 1

u fee the mark at F, and chere ſcrew it faſt. Obſerye the bearing of the compaſs*; which muſt, be entered in your field-book..

3 44%

Next move the index of the theodolite=

mark at B, and there note the angle FAB, cut upon the. \theadolite-part by. the nonius of its index; this you muſt alſo enter in

your field- bock. The inſtrument remaining

in the ſame poſition, let your aſſiſtant ere& a mark of equal height therewith, upon the ſummit of the firſt riſe of the ground at 2. Raiſe the elevation-index, until, through the

fights, you ſee the mark at and there ſerew it faſt, Meaſure the diſtance An,

and proceed to find the ſsegment of the baſe, &c. as directed in Chap. VI. This done, remove your inſtrument to 2. Place a mark at o, and depreſs the elevationindex, until, through the fights, you. per-

ceive the mark at o. Screw it faſt. Then

meaſure: the diſtance 70, and proceed to

The degrees, &c. to which the needle points; it being the angle that the line AF makes with the meridian,

find

find the ſe eat. as before directed, Con · tinue the proceſs in the ſame manner to B, Add the feverat ſsegments together, and the ſam wilt be the true length of the baſe-line AB; which enter in your f geld-book. Move the mark from B, and fix your infttument

Alfter part. to the 360th degree. Move the whole about, until, through the fights, you fee the mark at A, and there ſcrew it faſt, Move the index of the "rheodolfte-part, until, through the fights, you ſee the mark at C; there note the angle A BC; and meaſure the diſtance BC, as before directed. Proceed in this manner to meaſure the remaining fides and angles of the figure, and inſert them in your field-book.

It may be here neceſſary to obſerve, that the marks erected in the ſeveral angles of the field muſt be upright, and the inſtru-

ment, upon each removal, placed on the

fpot where the mark ſtood; for without this

precaution, an error will ariſe. ; N. B.

?

3. 88 a

F here the mark ſtood. Let the mark be teflaceſ at A; then ſet the index of the then

N. B. To prove the truth of your men- ſaration, multiply the nuinber of ſides encloſing the field by a; ſsubtract 4 from the product, and the remainder, multiplied by go, will be equal to the ſum of the ſupplements of all the external angles taken in meaſuring round the field. If the internal angles had been meaſured, the above rule Vould have given the ſum of thoſe angles themſelves.

+ 4 Fu £2 Wer Fe

Mo . 1

— 1 2

The Town FizLD or CoMMoNn.

— —

of - „ p

Angles. Deg. Min. Sides. Ch. Lin. Obſervations.

FAB 256 30 [ AB.|$8 80 [AB 210 30" Eaſt.

[ABC] 261 6 BC Ach. yo Un.

LA gate and ſtile J

/ BCD] 240 o CD 4.4 20 z ch. 5 lin, A brook. FR CDE| 250 o DE 5 50 Mr. Bottrel's fence. DEF 140 EF 3 10 l ch. 20 lin. A ſtile. EFA 292 30 [FA s Tus lence. WEN SO REM Tt : Þ 11 Eu — 4 , 1 qt J 245 Ib aeg lber N E CHAP

1 1 41 2 8 1 ; 1 1 22 4 ini ' SMT wont tO QNE17T I SYUI Oi — 5 A © v4 4 — , Me 741 : Y 231 0 rare O17 enn has

Fp 0 1

2 % þ , :

7 E T ABCDEFGA eee an irre- l gular field. Fix your inſtrument ver- 5 in ſome convenient part thereof, as at

O, from whence the angular points of the field are viſible. Set the index of the theodolite- part to the 360th degree. Turn the whole about, until needle points di-

„ — —

. . og ; Aa 2 r 1 Tg (af JV 3

1 Is SI

S nods

N en! to the north and ſouth of the compaſs ®, and there ſcrew it faſt. - Then direct the ſights - to, A. Note the degrees, Sc. cut upon the -tbeadelire-part by the nonius; and enter them in the ſecond; co lumn of your field book. This done, mea ſure O A, the diſtanee ſtom your ſtation to A; inſert it in the fourth, column of your; bock. I Then direct the to B. obſerving; the degrees, Scl cut by the, n nius as before: theſe enter an the ſecond: car lumn of, your field- book. Next meaſurg OB, the diſtance from your ſtation to B,. which inſert in the fourth column of your bogk. Point the fights to C, and proceed, as before, until the angles are all meaſured. But if there ſhould be a riſe or fall in the land, from your ſtation to any of the angles, as in the line O 7 q F. then muſt the baſe or

horizontal line be found, as directed in Chap. VIII.

The ſights will then form the meridian line NS.

$a .

KE 2 Directions

Birch, 1 far lere Hin the the former ji, Wir & your" pencil woch weridin übe Ns, and aſſume any point, as O; therein for your ſtation. Bay the diameter of the protractor upon the meridian lire N 8, with che center thereof ax O. With your pro- — point off the ſeveral degrees, Sc df the angles in your field-book From draw right lines, containing the angles marked off by your!/protrattor? With your voinpaſſes, from a ſeale of equal parts, ſet off {from your field- book) the diſtances O A, , Cc. meaſuted from your ſtution O. Join the ſeveral points, thus determined, until the whole be encloſed, and you will have the eee field.

fee

Y 8 1 OT SES Ae ns £ 8 2 7 TK F . dl ER eee: 5 .

by nt

a

Z

88 1 We would recommend the ule, an entire circle for protracking! wi 0

1 0 # # 3 111 2 #3 4 p — 33 44

Chap.

nt 53 1

7 0 wah an Fee feld with 4

chain alone.

r a Eo do os

L ABC DEA ee a level

but irregular- ſided field. Erect viſible marks in all the angular points, A, B, C, D, E. This done, run your chain from A in a direct line to B, alſo from B to C, and inſert their meaſure in the middle column of your field-book. From C, run your chain direly to your firſt ſtation A, and enter its meaſure in the field-book : thus will the triangle ABC be completed. Proceed again from A, and meaſure in a ſtraight line to E, for the crooked hedge AE: take off-ſets from the hedge in the ſeveral, crooks as you go along, and inſert them i in

r enn according to the form an- 5 E 3 nexed,

L 54 ] nexed. Then meaſure from E to C, from C to D; and laſtly from D to E: thus will

the proper requilites for protracting the field be obtained.

NEW CLosE. Of-ſets and Remarks. Sides Of-ſets and Remarks. a 9k Length. fk ; Lumley's hedge, 310. Into Crump's lane gate, 200 o BOG 8 600 4 Diagonal, — 720 A to E FP o00 ooo. Lloyd's hedge, 1 200 1070. 1 5 300 [020, td üb 440 [o 0 : E to C Diagonal, — — 500 7 IC to D Kerry*s lane, — '—f 220 SHi9i&7)) 6 a. 400 EA 1 81 a ( Nenn 164602 ene 5 0 CHAP

o-

ot. 55 +

e H A P. KI. $ 0 of the of the inſtrument i in de,

2 7T# a rr

H E RE are various methods for performing the buſineſs of levelling ; 3 but we think the following preferable, as 95

being the moſt certain and expeditious.

Having provided two ſtation- ſtakes, eight or ten feet in length, every foot thereof di-

vided into a hundred equal parts; to each + ſtaff muſt be fitted a vane about three

inches ſquare, to ſlide with a ſcrew, in the back part, by which it may be confined. at

any height upon the ſtaff, ©

Obes a ˙²ÜÜn tr Anois pon Lied 8 4; 5 n „ ry 7 Ee CR oo , PPP T LO LI IR 3 o 2 N mY 7 - — 1 I a x ay + IE Cs 8 124 U EFF , 2 N . n 1 þ JT MMK. ß 7˙% pf

Direct S.

F

Place your inſtrument in a right line, joining the two ſtations to be levelled, and fix it perpendicular to the horizon. Set the elevation-index to the center- diviſion of the

ſemi- circle, and there ſerew it faſt. The inſtrument thus rectified, look through the

fights," and cauſe one whe: ur afſiſtants (pro-

en 589 $5193 14

eee,

a black line drawn ht acroſs: he middle.

E 4 . vided

£ —

vided for chat purpoſe) to ſlide the vane

be cut by the Horizontal wires, noting the diviſions upon the ſtaff from the ground to the Kid black line. Then remove to the

other fide of your "inſtrument, and look through the fights, canſing your , ther aſ- —5 at the ſame time, to move the vane pon the other ſtaff, until the black line 3 K horizontal wires coincide." ; Note the diviſions upon the ſtaff, from the ground to the ſaid black line. In conſe- quence of theſe obſeryations, if the ground be level, the number of the diviſions cut by the vanes on both ſtakes will be equal. On the other hand, if thoſe numbers cut by the vanes be unequal, their difference will ſhew how much the ground declines from a true level.

. here ohe, inter vene.

1 che places to be levelled have inter-

vening obſtacles, or ſhould be too far diant from each other for one. ee

1414 by 390303] 3. 1

upon the aff, until the black line thereon

©

2 o - # #® l " Y O f " x : s $4 I 4844 +4 —— — td as Gf Ss 4 y % . Ia a4

"

Es] Yt 7% 3"

fs. 74 F g 4 ; ot 1 $8

E us

It Ay

Fe

1 þ

and keep a regular account of the numbers

cut upon the ſtakes: let thoſe of the back

ſtation be put in the firſt column, and thoſe

of the fore ſtation in the third, obſerving to regiſter the number of ſtations in the in-

termediate column. As for example:

a MES. XY "0-0 4 : .

Back Stations, Stations. Forward Stations.

O 29 1 1 li [435 32; 2 F 4% 0 4 4 — 1 1. . 2 45 5 he. NE + vt 7S-" yy 3 2 95 J 4.0 27 J#<4,248 34.

Thus you may determine not only the acclivity or declivity of the ground at every ſeparate ſtation, but alſo of the whole

together. For the difference between the

4 Ww + 2 10

p The hd aff will become the. backward af at new obſervation, 2 reſpective.

you muſt take as few ſtations " as poſſible,

1 8] reſpective ſum of the forward and the backward ſtations gives the aſcent or deſcent re- Note, In levelling rivers, the firſt and laſt ſtation muſt be taken cloſe to the water's edge. But the intermediate ſtations may be taken, at pleaſure, in the meadows or fields adjoining 8-

PT TTILITD CH AP. XII. x „ th We f the 1 in W

To find the a point for opening 4 drain.

IX your inſtrument in a 1

lar poſition at A, the head of the

ſhaft of the mine; erect a mark of the

' fame height therewith at the foot of the

kill B. Depreſs the elevation- index, un-

til;

% :

o „ All 1 r I — ; qe ages he Sas abt 2 15

til / thro the ſights, you ſee tlie mark at B, and there ſcrew it faſt. Then the depth of the haft AC being known, and the altimeter fixed perpendicular to the horizon,

move the radius as occaſion requires, until the aforeſaid depth AC, be cut upon the

altimeter by the elevation-index. This done,

you will find the true length of the drain

from the bottom of the ſhaft C to D, „

cut by the inſide of the diameter on the radius, , and alſo the diſtance from the ſhafthead A, to the part D on the fide of the hill,

where the drain muſt be opened, cut upon

the elevation-index by the perpendicular line of the altimeter. 7

Chap Xiii.

A certain method of finding which way the LOAD of @ mine inclines, in order to fink a

2 new ſhaft. E ING provided with two Nations ) ſtakes, about four feet high, with

a ſocket in the top wherein a candle

+ +4 4

TE may be fixed; place your 5 directly in the center of the ſhaft, and one of the ſtakes, with a lighted candle therein, in the firſt angle or turning of the drift, at ſides of the mine. Then lay the indar of the t heodalite- part to the 360th degree: turn the whole about, until the needle hangs directly over the north and ſouth points of the compaſs, and there ſcrew it faſt. This done, move the index of the theodolite- part, until, through the ſights, - you ſre the lighted candle, and then obſerve the degrees, &c. cut upon the heater. part by the nonius of it's index, which put down in the ſecond column of your memorandum- book, according to the form annexed. With your tape-hne meaſure the diſtance from the center of your inſtrument to the candle, and enter it in the third column of your book. No- tranſpoſe both the inſtrument and ſtaff, fixing the former perpendicularly on the ſpot where the ſtaff was, and the latter where the inſtrument ſtood. Ereck the other in the! next m_— of the drift, in the 0 manner

ans as above directed. Bay ti ddös of the theodokte-part as before, turning the whole about, until you fee the candle at the

fult tation; and there ſore w. it faſt. Move

the ander of the I headblite-purt, until you ſes:the candle in the ſecond ſtation; observe

the degrees cut by the nonius of! its index,

and enter it 6 in the ſecond column of L your

bock. Meaſure the diſtance from your aul ſtrument to the ſecond candle, which enter in the third column of your: boo

No change the ſtations of the inſtru ment and ſtakes: the former muſt be faxed in the ſpot where the ſecond candle ſtood,

and that muſt be removed to the next angle

of. the drift, while the firſt muſt be placed where the inſtrument was laſt erected. This done, proceed as before directed, until the whole be finiſhed. The meaſurement in tbr mine being taken, you. muſt take the ſarne above-ground in the following manner.

Cover the mouth of the ſhaft with boards,

and place your inſtrument perpendicularly in the center thereof. Then thei;ndex of the

Ane ach ni thealloliter

; ' 7 , T +

f

1 62 J thtodoliti+partibeing fixed to the 360th. des gree, turn the whole about, until the needle ſtands directly over the north and ſouth points of the compaſs, and there confine it: Move the index of the beodalite- part to the aumber of degrees, Gc. of the firſt {tation taken in the mine, and entered in your mo- ſtrument remaining in the ſame ' poſition, let your aſſiſtant erect a mark of equal height therewith, in a direct line with the lights, about a yard farther than the diſtance of the fi meaſure in your book. Then raiſe or depreſs the elevation-index, until, thro

the fights, you ſee the ſaid mark; and hav-

ing ſcrewed it faſt, (the a/timerer at the ſame time being confined in a perpendicular direction) move the radius until the ſaid /f meaſure be cut thereon by the inſide of the

land, from the inſtrument to the next Nation, ci cut upon the elevation- index by

the perpendicular line of the altimeter,

which enter in the fourth column of

oc wt 0 4. @ En

Zameter. Vou will then find the diſtance to. be meaſured upon the ſurface of the

— td. — te OE. ee Cots IS

TI 63 J your book. Vou will find falſo the perpendicular-aſcent or deſcent" of the ground in that diſtance, eut upon the altimeter by the elevation-index;" and this muſt be entered, if an acclivity,' in the fifth ; if a declivity, in the fifth column of your bock. Then with your tape-line meaſure the di- ſtance 43-infected in che fourth column of your book w. from your ſtation towards the mark. At che Meridia 6f thi is liztied fix your inſtrument. 2113

'Let your aſſiſtant remoye the firſt mark

to the ſpot on which the inſtrument ſtood. Set the inder of the rbeodblite-purt te to the 360th degree." 2 Turn the whole : about, Fn you fee the aforefaid. mark i in its new ſtation, and there ſcrew it faſt. Move the index of the rheodolite-part to the 81 of degrees, Gc. inſerted in your book the ſecond ſtation under ground. Cauſe your aſſiſtant to erect another mark, about a yard beyond the diſtance entered in your book af the ſecond ſtation, as before directed. Raiſe or depreſs the elevation index, until, thro' the lights, you ſee the ſaid mark; there ſcrew it he cates faſt,

wy

Fe &

Ea of 6.

Titel:

faſt, Proceed now in all reſpects as before laid down, until the whole be acoompliſhed. Then the difference between:the xeſpeCtive famsof theaſcents and deſcents (being added to, or ſuhtracted from; the depth ef the main Maft) gives the depth of the ſhaft required, according. as the ſum af the aſgents. was greater or leſs than chat of the deſcents: the pot. thereferc, where; the laſt męaſurement

coxminated, will, he che direct place t the new ſhaft muſt be pened j

O Sos 100

inr 1 fr

Pr 110 704

09K.

N 2 2 82

WN 3 in the Centeg of —

vun oi at uon

ur panes

10 —

I þ "J Mey...

22 — .

—

1 % 4

ro 241 u K Surghod nom 340

u

2 WW U mM f 4 5

1 8

ö

—— —

er

18 at} 0 55

—

men 1

— —

Flo 4 ,

— —

FA 4 4 uf

L.A n

—

m

nm

— —

—

-

+ 4

;

: :

—

—— —

Aj (©

ty

2 Sts: -

#% as X A ! - il 4 : 70199 © A 1 Sc. ; fa £S £ p Bo-2 ph e 7 „ 4 + Ls b 3 J s ;

1

Oe Ha P. Xiv.

% 7% I.

CAVIN G already deſcribed the uſes. of our inſtrument, (with regard to

the ade of ſtanding timber, ſurveying, Ge.) as far as the nature of the preſent work would admit; yet, perhaps, it may not be improper to ſubjoin ſome entertaining examples, which may farther ſerve to evince it's utility in ecfunning trigonometrical N

Py a To 117 #

To meaſure the breadth of a river.

Let D E repreſent a river, hate breadth is required. Place your inſtrument at any al 8909, 3 convenient

convenient point, as at B. Lay the inder of the theodolite-part to the 360th degree. Turn the infirument about, until, through tho fights, you ſee ſome particular aſimed point upon the object A; and ferew it faſt, This done, move the index to the 270th degtee l. Cauſe your aſſiſtant to erect a mark, in a right line with the fights, at any ckſtance, as at C. Tranſpoſt boch the in- ſtrument and mark. Meaſute ow" diſtance CB, equal to 200 yards. 7 1 Now lay the index of the eee to the 360th" degree, as before. Turn the whole about, until, through the ſights, you ſee the aforeſaid ume point upon the object A; and ſcrew it faſt. Move the index, until, through. the ſights, you. ſee the mark B, and obſerve the quantity of the angle AC B; which admit to be 360 84. Then the — wi 8

As the co ine or © 8 - 9. 903 11 Is to GR - n 2. 30103 So is ſine of £ 8 36 52 9.77812 To BA re - 150 yards, 2.17604 +41 being at right A with the 360th degree.

Not Cs

Note, If, in the foregoing proceſs, the dex of the rheodolite-part were laid to the goth degree, inſtead of the 270th, turn the inſtrument about, at your ſation C, until you ſee the mark B. Then take the angle B C A, and proceed, in all other re- ſpects, as above directed.

Note alſo, If you cannot fix your in- ſtrument cloſe to the river, you ſhould meaſure the diſtance from your ſtation B to the edge of the water; which, ſub-

tracted from the whole line BA, gives the

true breadth of the river required.

To work the above proportion upon the in-

ftrument.

Set the elevation-index to 36 52 upon the ſemi-circle of the inſtrument, and ſcrew it faſt. Confine the altimeter in a perpendicular poſition, and move the radius until

the diſtance (CB 200 yards) be cut upon it

by the inſide of the diameter. Then you

The quantity of the angle cut 5 the nonius on f the ſheoublila. part.

F 2 will

— —— — by

al find the breadth of the river (BA 150 1 yards) cut upon the altimeter by the elevation index; and alſo the diſtance. (CA 250 5

yards) cut upon the elevation-index by the perpendicular line of the altimeter.

" 2 4

:

+ $

:

E

Fo

F . : b Ty -

bs

[6 3

7. Chap. Xv.

7 0 Sw the dr jbance between Py rene te er, x ; : k 8 '%

NT 7 g 2 ny) N L . . - Dx % — . x . 8 - wa — e. — nn D 23 — hy

e —

remote objects, whoſe diſtance. from the ſtations B, C, and likewiſe from each other, is required. Place your inſtrument at C, and a mark at B. Lay the index of the theodolite-part to the ey degree. Turn the inſtrument

F 3 „„

UPPOSE A and D to repreſent two

r

— .

% . — 4 2 — 5 1 — 2: ate — 4 „ R n ad — — — — P ˙*ꝛ-ů̈uà EE. TERS 72

—

4 p

about, until you ſee the mark at B, and ſcrew it faſt." Then move the index, until vou ſee the object A. Obſerve the quan- GE Wen angle B C A; which admit to be 51 26, Move the inder until you ſee the object D, and obſerve the quantity of the angle BCD; ſuppoſe 120% ©. Now tranſpoſe both the inftentaigt and mark. Meaſure the distance B C (equal to des yards), Lay the index of the t heodolite- Part, as before, to the 360th degree. Tum the inſtrument about, until you ſee the obect A, and ſcrew it faſt. Move the index, until you fee the abject D. Note the quantity of the angle ABD (equal to yo* 0.) This done, move the index, until you ſee the mark at C; and note the quantity of the angle ABC, ſuppoſe 105* . Your obferyatians thus completed, proceed

To work the ſeveral proportions upon the i in- Hrument.

Set tke angle ABC (105*) on the small ſemi-circle of the gltimeter, to the nonius upon the radius, and ſcrew it faſt. Then fet the nonius of the elevation-index to the

angle

angle BOCA 51 26 upon the tiki-Lioald 5; of the inſtrument. This done, move the radius," until the diſtance CB (966 yards) be cut thereon by the inſide of the diameter. You will then find the diſtance BA 1566 yards, or one mile, cut upon the altimeter by the elevation-index ; and allo the diſtance CA='2171 yards, or 1 m. 1f. 34.22 p. cut upon the elevation-mdex by the een

lar line of the altimeter. © By a similar proceſs the medlirement af the ſides of the remaining will " found, a8 follows; n

1 ' Yards. © M. F. P, . Re Res BD 1853, or 11 164 42. 1 CD 1223, oro 5. a a AD 2073. or I 41 16 mh 5

It may here be neceſſary (6 revhark that in the application of our inſtrument to meaſuring diſtant objects, we been led to the following conſiderations. ay I an angle be taken ou? of 95 1 of of the horizon, ſuch angle is leſs than the horizontal one ſubtending the ſame, - ar

5 equal,

equal, ; fide., From an obſervation, therefore, taken out of the horizontal plane, no true plan can be formed, nor until it be reduced. thereto ;- Which . be fected thus. 228

Set the zonius x" the . upon

ſemi-circle of the inſtrument, to the degrees, Gc. in the angle made by the in-

terſection of the plane of the obſerved angle, and the plane of the horizon. There ſcrew it faſt... Confine. the altimeter in a perpendicular poſition. Then move the radius, until the baſe- line of the obſerved angle be cut upon the elæpation-index by the perpendicular line of the altimeter, This done, you will find the correſponding horizontal

baſe-line cut by the infide of the diameter Per the radius.

Nou ſet the degrees, Gt. contained be- n baſe and ſide ſubtended by the Db ſerved angle, upon the ſmall ſemi- circle of the altimeter, to the nonius of the radius, and ſerew it faſt, Then ſet the elevarionnder tò the number upon the altimeter, viz. the length of the hs l the obſerved dung 5 angle,

angle, upon the altimeter. This done, you will find the correſponding horizontal angle upon the ſemi- circle of the — cut by the noniur of the elevation inder.

Note, If the different objects A different angles of inclination to the plane of the horizon, they muſt be considered repectirely. eee UL OS

It is requiſite to obſerve, that, in very

remote distances, the horizontal lines ought to be conſidered as tangents to an arc of the earth's circumference. To find the an gle, at the earth's center, ſubtending the ſaid tangent, we have the following rule: 2

As the ſemi-diameter of the earth i is to the horizontal diſtance meaſured ; ſo is the radius (go*) to the tangent of the arc required.

Now to find che length of the ſaid arc, multiply the earth's circumference by the quantity of the angle ſubtending the tangent : the Por by 3605 and che

In 3 operation your multiplier a Hiviſas mins always be of one and the ſame denomination z whether degrees, minutes, &c.

quotient

[ 74 ]

quotient will be the length of the are;

which i is the true line to be delineated.

P

eee.

Set Ice 6 Chap. Xvi.

eg

fo fd the beight and diftante + 8 eee or battery, #'s elevation or depreſſion

above or below the horizon, when the . ground between the stations is not in the ſame borizautal plane.

ns o I

A

Ine.

W P Mir

Jjcip

Wo WJ )

Ww Wy

&

— e Ne

" aa

N 8 N

E AC repreſent a tower, and CDEF an adjacent hill. From the ſtations D and E it is required to determine the horizontal distances D 7. En; the extreme

diſtances

[ 1

diſtances D A, DC, EA, EC; altitudes gh,

1A; and the height of the tower AC.

Place your inſtrument at D, and meaſure

the angles qDC, DA. Turn your in-

ſtrument about, and meaſure the angle

BDE. The ſum of this angle, and the former 9 DA, being ſubtracted from 180, leaves the angle ADE.

This done, tranſpoſe the instrument to E. Meaſure the diſtance DE, and take the angles of depreſſion 2E C, zED, and angle

of elevation E A. The ſum of the angles nED, nEA, gives the angle DEA; to

which add the angle ADE, and ſsubtract the ſum from 180*, the remainder will be the quantity of the angle DAE.

Fr Om theſe obſervations the ſeveral parts, i

as required in the above problem, may be found by the following calculation.

As the fine of . 4 DAE Is to distance 3 So is fine of - L DEA Io the viſual line DA

2. A

— — — 2 8

JJC 2 Is to EK e nn DA

80 is fine of ß ([ DA alan: TO ien bone „ gle; . And ſo is eo- ſine 1 . 7DA 3 St f D.

n A : Oar ag , a Eo e YE OWE os mn OR COS .

As co-ſine zZ. DC

an 10 n 7D So zs ſine 2 om 4qDC And ſo is radius EE. or ET e

e As radivis 7 . Is to diſtance N So is co-ſine BDE — To a — 2 DR: And ſo is fine e BDE To — „

As co-ſine 4 EC 1 - 1 E. So is radius 3 To viſual diſtance CE "I Thus a

t ws

Thus the diſtance of any propoſed point of A tower, or battery, either above or below the horizon, may be found by directing the obſervations to that particular, propoſed, point. Note, The above proportions may be performed upon the inſtrument, according to the directions 1 in the et oper: 0

A

4 : vs © bs .

$

uu! CHAP. XVII.

To meaſure the fron of a houſe.

THE H EIO Hr.

. ACE your inſtrument towards the left ſide of the houſe, at right an gles with the corner. - Confine the altimeter in a perpendicular poſition. Meaſure your diſtance from the houſe to the axis of the inſtrument. Move the radius, until that diſtance be cut thereon by the inſide of the diameter. This done, depreſs the elevationindex, until, through the ſights, you ſee the bottom of the houſe: there note the number cut upon the altimeter by the ſaid index. Then raiſe the elevation-index until TY you ſee the top of the houſe, and there note

ESATU AT the

— —

tat

the number cut upon the altimeter by the

aur the ſum of theſe numbers is the Berght of the houſe required:

TAE BRTA DTH. Lay the Borivonral. indbæ upon the fir division of the moveable limb of the hiding. piece. Obſerve, through the fights, that

the left edge of the front be cut by the

perpendicular wires. Set the moveable limb of the ſliving-piece parallel to the front. This done, move the index alone, until you ſee the edge of the houſe, on the right hand, cut alſo by the perpendicular wires: you will then find the required breadth expreſſed upon the moveable limb of the Hiding- piece by the horizontal-index.

In like manner the height and breadth

of 4 window, in the front or fide of a houſe,

285 be ality obtained. , To theſe examples already given, many others, we apprehend, will readily occur to the reader; whereby the extenſive

uſe of this inſtrument might be farther illuſtr ated, ;

Tables

A ER Ne eee Ines II HR

Sp. E Ned -1.X

HE principles of geometry, con- : 4 firmed by obſervation, ſhew, that N angles either acute or obtuſe, taken by the inſtrument, when out of the horizontal plane, are very different from their correſponding or projected angles upon the plane of the horizon; ariſing from the different ſsituations of the obſerved objects: © VIZ. n

Firſt, If the objects be elevated above, or depreſſed below, the horizon, the obſerved angle will be 1% than its projected angle upon the horizontal plane.

Secondly, If one of the objects be elevated or depreſſed, and the other zz the plane of the horizon, the obſerved angle will. be greater than its projected angle upon the horizontal plane; except the obſerved angle

8 be

be 90? oo 7 for, in ſuch caſe, the obſerved and projected angles will be equal. Thirdly, If one object be elevated and the other depreſſed from the horizon, the

obſerved angle will be greater, or leſs, than

its projected angle upon the horizontal plane, according to the difference of the angles obſerved upon the ſemi- circle of the inſtrument, and ſmall circular arches.

I. If the angle, obſerved upon the ſemicircle of the inſtrument, be leſs than the angle obſerved upon the ſmall circular

'arches, the obſerved angle upon the ſmall

quadrant (fixed for that purpoſe upon the horizontal index) will be greater than its projected angle upon the horizontal plane.

2. If the angle, obſerved upon the ſemicircle of the inſtrument, be greater than the angle obſerved upon the ſmall circular arches, the obſerved: angle upon the ſmall quadrant will be leſs than its projected angle upon the horizontal plane.

N. B. If the angles obſeyed upon the ſemi- circle of the inſtrument and ſmall cir- : cular

- 81 1 5 55 cular arches be alike (or the ſame) the angle obſerved upon the ſmall quadrant, and its projected angle upon the plane of the horizon, will alſo be alike, or the ſame.

In conſequence of the addition of a ſal! quadrant upon the horizontal index, (ſee Deſcription, page 14*) it will be neceſſary to give a rule for reſolving the following

Pr Ob L E M.

The meaſure of an angle being taken either above or below the horizon, to find the quantity of the correſponding or projected angle upon the horizontal plane.

The RU LE.

The altimeter being in a vertical poſition, ſet the nonius of the elevation index to the degrees, &c. contained between the plane of the obſerved angle and the plane of the horizon, and there ſcrew it faſt. Move the horizontal index, until the angle obſerved upon the limb of the theodolite be cut upon the ſmall quadrant by its nonius.

G 2 Note

Note the number cut upon the moveable limb of the ſliving- piece by the horizontal index. This done, ſet the elevation index

to oo degrees, and screw it faſt. Set the Borixontal index to the ſame number (as

above noted) upon the moveable limb of the ſliving-piece. You will then find the

- correſponding angle cut by its nonius upon the quadrant.

Fo meaſure an angle which has different zmclinations to the plane of the horizon.

KR. Lk:

Ty the horizontal index to the firſt diviſion

on the moveable limb of the ſliving- piece. Turn the inſtrument about, until you ſee the object towards the left hand. Move the horizontal index, until you ſee the other object towards the right hand. Obſerve the number cut upon the moveable limb of the ſliving- piece by the ſaid index, and alſo

the degrees, &c. upon the ſmall circular

arches.

This

5 3 a Ci Fay

„ This done, the obſerved angle may be reduced to the plane of the horizon by the following

Set the elevation index to the degrees, &c. obſerved upon the ſmall circular arches. Note the number cut upon the radius. Move the radius until the number obſerved

upon the moveable limb of the ſliving- piece

be cut upon the elevation index by the

perpendicular line of the allimeter, and note

the number then cut upon the radius. Now fet the elevation index and ſmall circular arches to oo degrees, and draw the radius to the number firſt noted thereon. Lay the horizontal index to the fame number upon

the moveable limb of the ſliving- piece as was obſerved the ſecond time upon' the radius: then will the correſponding hori-

zontal angle be pointed out by the nonius

upon the limb of the quadrant.

At page 18, add the following method for ſetting the altimeter in the ſame direction with the tree:

Before

—

84%

Before the horizontal diſtance is meaſured

/

with the tape-line, Sec. take the inſtrument b in your hand, from your ſation, either to t the right or left ſide of the tree, as may be ; moſt acceſſible. Hold it ſide-ways, at any,

convenient diſtance therefrom, in a vertical n

poſition, obſerving to keep the diameter of the inſtrument always on the fide next your 7 ſtation, Then move the altimeter, until you obſerve it form the ſame angle with the radius, as the tree forms with the horizon; or until the lines of columniation coincide.

At page 19 When the radius muſt be moved to cut the diſtance meaſured from the tree, it is proper to obſerve, that if you cannot fix your ſtation within the diſtance of 20 feet from the tree (it being the number of equal diviſions upon the radius, or if the length thereof ſhould extend above 40 feet (the number of equal diviſions upon the alfimeter,) in ſuch caſe you muſt move the radius to 2, 4, or 4 of the diſtance meaſured. Then the height and diameter of the tree will accordingly be

r tus Swat fo ond an oa mm.. Sk.

o

be twice, thrice, or four times as much as the height and diameter meaſured upon the inſtrument. Otherwiſe you may call the diviſions by any other denomination, as yards, poles, furlongs, miles, &c. agreeable to the denomination in which you take your diſtance.

To adjuſt the inſtrument. EUR

The inſtrument being fixed upon the ſtaff, flacken the ſcrew, in the parallel plates, on the fide elevated; at the ſame time ſtreighten the oppoſite ſcrew. Proceed

thus until the whole inſtrument may be -- turned round upon the ſtaff without altering g the due poſition of the bubble in the ſpirit-

level. In the ſame manner the dendrometer may be ſet vertical to the plane of the horizon by the adjuſting ſcrew, in the part by which it is fixed to the FBeodolite.

N. B. If the plane of the limb of the 5 theodolite be ſet parallel to the plane of the

34S

1 86 ] the horizon, and the dendrometer ſet exactly vertical, ſo that it may be turned round without altering the true poſition of the plummet; it will then follow, that if the limb of the f heodolite be exactly horizontal, the dendrometer will be exactly vertical.

Er R A T A.

R L At page 17. True Solidity 7 26 114 Com. Solidity 5 44 62

Tables.