The New-Jersey gazette — Burlington [N.J.], 1778-08-26 00:00:00, page 1

To the Printer of the New-Jersey Gazette.
S I R,
SOMETIME ago I had occasion to extract
the cube-root to four or five places of fi
gures, but could not recollect the common
methods for doing it; and not having lo
garithms at hand, I fat down to make a
rule to answer the present exigency : This
led me to confider the fubjed much further than I at
firft intended, and produced the following theorems,
which may be useful to the mathematician ; there
fore their publication in your paper will greatly
oblige EXPONENT.
fr d ") „;— Refelvend given.
1 ' ''°' r =Root assumed.
^ 2rr e —Part fought.
^§: r IR— Root found, and
^^ 12M4-r3 I d=Difference between
<^ - —=R „
I the refolvend and r'
—+2rr;
These theorems converge faft, and may serve very
well for common use, when no great exa&nefs is re
quired at one operation.
The third theo- ^|i
rem is, 2:— + r
r - =:R for the cube root.
3
2: — +2r
- —-— —R for the 4th power.
4
2 — + 3r
—^ —=R for the sth pow
-3 5 er. &C.&C.
These expreflions always give the value of R too
much, if not exadly true; but as the excess is in a
certain ratio to the true root and root assumed, it
may be determined to 10 or 1% places of figures by
the second and third theorems: Thus in the cube
root, let x— the excess, then
Qrr-zr:^
r , mb
- —x. which sub- 2: -f-r
m \ T ftra&ed from r
s'+4—-5= 3
gives the root true to —- °' a "
100000000000 unit.
To fihd the value of jx in the divisor, multiply as
many of firft figures in the quotient by 5 as there are
correspondent figures to the left hand, between
—+ rr, and 2r: ^l°
r r
But universally, the value of e may be found to fix,
eight, or ten places of figures, by the firft and third
theorems, thus ;
. =r in the cube-root.
— + iorr-| -i6r: "^
r r
in the sth power.
2 —+7, +i6rr: ? " ' T .
r r
m 6 ^c. &c.
5 —4-12, +32^^
The roots of all even powers are belt found by the
square and cube-roots.
Notwithstanding these theorems consist of pretty
many parts, yet the sameness of these parts renders
them very easy to work, as will be found on trial;
especially as r may be assumed a single figure with
cyphers, and as the operation needs not to be re
peated,
These theorems may, I believe, be applied to mixt
er adjuted powers. The general expreffiong for
42 22
& 4-R —^, or R' +?;R -fR — m, &c. &c.
2 — —-for 2 ——zr—i| 4-r=R.
will be ,—— —. and —-
3
But a disagreeable situation, and want of health, pre
^^ my ptirfuing the fubje& any further at present.
NEW-JERSEY GAZETTE.
[Vol. I.]
WEDNESDAY, August 26, 1778.
In CONGRESS, May 27, 177^
ESTABLISHMENT of the American Army.
I. Infantry.
RESOLVED, That each battalion of infantry
shall consist of nine companies, one of which
shall be of light infantry ; the light infantry to be
kept complete by drafts from the battalion, and or
ganized during the campaign into corps of light in
fantry :
That the battalion of infantry consist of
^f I Colonel and Captain, 75 dollars per month.
1 I Lieut. Col. and Captain, 60
° I I Major' and Captain, 50
\ 6 Captains, each - - 40
p: I Captain Lieutenant, - 26 2-3
0I 8 Lieutenants, each r 26 2-3
0 IgEnfigns, each - - 20
Paymaster, ) . (20) In addition to
Adjutant, , >'° Al 3 /theirpay as offi-
Quartermaster, ) "° mtn ""°(13 Jeers in the line.
I Surgeon, - - 60
i Surgeon's Mate, - - 40
I Serjeant Major, - - 10
1 Quartermaster Serjeant, 10
27 Serjeants, each - - 10
i Drum Major, - - 9
i Fife Major, - * 9
18 Drums and Fifes, each 7 1-3
27 Corporals, each - - 7 1-3
477 Privates, each - -6 2-3
Each of the field officers to command a company.
The Lieutenant of the Colonel's company to have
the rank of Captain Lieutenant.
11. Artillery.
That a battalion of artillery consist of
r 1 Colonel, - - 100 dollars per month.
I I Lieutenant Colonel, - 75
0 ; r M^jnr, - - 63 1-2
{i2 Captains, each - - 50
: 12 Capt. Lieutenants, each 33 1-3
I! 12 First Lieutenants, each 33 1-3
0 (36 Second Lieutents. each 33 1-3
Paymaster, ) T o be taken (25) In addition to
Adjutant, [fromtheline. , r s [ their pay as offi.
Quartermaster,) (16)cers in the line.
I Surgeon, - - 75
I Surgeon's Mate, - - 50
I Serjeant Major, - - 11 23-goths.
I Quartermaster Serjeant, 11 23-goths.
I Fife Major, - - 10 38-goths.
i Drum Major, - - 10 gS-goths.
72 Serjeants, each - - 10
72 Bombardiers, each - 9
72 Corporals, each • - 9
72 Gunners, each - - 8 2-3
24 Drums and Fifes, each 8 2-3
336 Matroffes, each - - 8 1-3
111. Cavalry.
,That a battalion of cavalry consist of
^f I Colonel, - - 93 3-4ths dollars per
o ; i Lieutenant Colonel, 75 (month,
^j I Major, - - 60
'g I 6 Captains, each - 50
I! 12 Lieutenants, each 33 1~3
q t 6Cornets, each - 26 2-3
1 Riding Master, - 33 1-3
Paymaster, ) ( 2 5) l n addition to
Adjutant >To be taken J IS C their pay as offi -
Quartermaster, ) fr° m theline. ( 15 ) cers j n the line.
I Surgeon, - - 60
I Surgeon's Mate, - 40
I Saddler, - - 10
I Trumpet Major, - 11
6 Farriers, each - - 10
6 Quartermaft. Serjts. each 15
6 Trumpeters, each - 10
12 Serjeants, each - 15
go Corporals, each - io
324 Dragoons, each - 8 1-3
IV. Provost.
Refolded, That a Proveft be eltablifhed, to consist of
1 Captain of Provosts, - 50 dollars per month.
4 Lieutenants, each - 33 1-3
i Clerk, ... 33 1-3
I Quartermaster Serjeant, 15
2 Trumpeters, each - 10
2 Serjeants, each - 15
5 Corporals, each - 10
43 Provosts or Privates, each 8 1-3
4 Executioners, each - 10
This corps to be mounted on horse-back, and arm
ed and accoutred as light dragoons.
Refolded, That in the Engineering department
three companies be established, each to consist of
THE
I Captain, - - 50 dollars per month,
3 Lieutenants, each - 33 1-3
4 Serjeants, each - 10
4 Corporals, each - 9
60 Privates, each - 8 1-3
These companies to be inftru&ed in the fabrication
of field works, as far as relates to the manual and me
chanical part. Their business shall be to inftrud the
fatigue parties to do their duty with celerity and ex
aftnefs ; to repair injuries done to the works by the
enemy's fire, and to prosecute works in the face of it.
The commissioned officers to be Skilled in the neces
sary branches of the mathematics ; the non-commis
sioned officers to write a good hand.
Refolded, That the Adjutant and Quartermaster of
a regiment be nominated by the field officers out of
the Subalterns, and presented to the Commander in
Chief or the commander in a separate department for
approbation ; and that, being approved of, they shall
receive from him a warrant agreeable to such nomi
nation :
That the Paymaster of a regiment be chosen by the
officers of the regiment out of the Captains or Subal
terns, and appointed by warrant as above; the offi
cers are to rifque their pay in his hands ; the Pay
matters to have the charge of the eloathing, and to
distribute the fame.
Refolded, That the Brigade Major be appointed
as heretofore by the Commander in Chief, or com
mander in a separate department, out of the Captains
in the brigade to which he shall be appointed :
That the Brigade Quartermaster be appointed by
the Quartermaster General, out of the Captains or
Subalterns in the brigade to which he shall be ap
pointed.
Refolded, That two Aids-de-Camp be allowed to
each Major General, who shall for the future ap
point them out of the Captains or Subalterns.
Refolded, That in addition to their pay as officers
in the line, there be allowed to
An Aid-de-Camp, - 24 dollars per month-
Brigade Major, - 24
Brigade Quartermaster, 15
Refolded, That when any of the Staff officers ap
pointed from the line are promoted above the ranks
in the line out of which they are refpedively appoint
able, their ftaff appointments shall thereupon be va
cated :
The present Aids-de-^amp and Brigade Majors to
receive their present pay and rations.
Refolded, That Aids-de-Camp, Brigade Majors and
Brigade Quartermasters, heretofore appointed from
the line, shall hold their present ranks, and be admis
sible into the line again in the fame rank they held
when taken from the line ; provided that ne Aid,
Brigade Major or Quartermaster shall have the com
mand of any officers who commanded him while in
the line.
Refoh>ed, That whenever the Adjutant General shall
be appointed from the lite, he may continue to hold
his rank and commission in the line.
Refolded, That when supernumerary Lieutenants
are continued under this arrangement of the battali
ons, who are to do the duty of Ensigns, they shall be
entitled to hold their rank, and to receive the pay such
rank entitled them to receive.
Reforced, That no more Colonels be appointed in
the infantry ; but where any such commiffon is or
shall become vacant, the battalion shall be command
ed by a Lieutenant Colonel, who shall be allowed the
fame pay as is now granted to a Colonel of infantry,
and shall rife in promotion from that to the rank of
Brigadier; and such battalion shall have only two
field officers, viz. a Lieutenant Colonel and Major,
but it shall have an additional Captain.
May 29.
Refolded, That no persons hereafter appointed upon
the civil ftaff of the army shall hold or be entitled to
any rank in the army, by virtue of such ftaff appoint
ment.
June 2.
Refolded, That the officers herein after mentioned
be entitled to draw one ration a day, and no more ;
that where they shall not draw such ration, they shall
not be allowed any eompenfation in lieu thereof.
And to the end that they may be enabled to live in
a manner becoming their stations,
Refolded, That the following sums be paid to them
monthly for their subsistence, viz. To every Colonel
50 dollars per month ; to every Lieutenant Colonel
40 dollars per month ; to every Major 30 dollars ; to
every Captain 20 dollars ; to every Lieutenant and
Ensign io dollars; to every regimental Surgeon 30
dollars ; to every regimental Surgeon's Mate 10 dol
lars ; to every Chaplain of a brigade 50 dollars.
Refold, That subsistence money be allowed to
[Numb. 38.]