DEC 
give in a monthly flatement of the “debts and credits” of 
his men; and it is the duty of every commanding officer to 
examine each lift, and to fee that no injultice or esueaty 
has been countenanced or overlooked in fo important an ob- 
jet, as every a are between officer and foldier moft 
unqueitionably i 
DECA, in Geography , ariver of Spain, which runs into 
the Xalon, two leagues below Anza in n Aragon 
DECACHORDON, in Ania a ngeal inftrument, 
of ten ftrings, called by rews hafur, refembling our 
alae = : triangular igure, with a hollow belly, and found- 
ing fr e lower par 
DECACTIS, in Natural Hiftory, a name given by fome 
toa kind of ftar-fith, of the branched, or aftrophyte kind, 
whofe rays are ten in number, where they firft part from the 
body, and each foon branches out into a number more. 
DECADARCHUS, Acxaduexes, among the Greeks, a 
commander of a party of ten men. 
DECADE, a word ufed by fome old writers for the 
number ten; and decades, for an enumeration by tens: the 
word is formed from the Latin decas, which is derived from 
reek word of the fame import. The word has been more 
siege appropriated to the caper of books, g. d. decades 
o which the Roman Titus Livius is divided. 
Hence alfo came decadal caihmcie: the Decameron of Boc- 
caccio, &c. 
DECAGON, a plain figure in geometry, having ten fides 
and angles. If all the fides and angles be equal, it is called 
a regular decagon, and may be infcribed in a circle ; for the 
method of sg which, fee PENTAGON. : aged, 
4.) be the fide of a regular deca in- 
€ a mean proportional between the fide of - gon 
and ane fum of t de and the radiu 4 B: 
A tA Produce A B to 'F, a ies 
F n may be equal to AD, and draw DF and DB, 
the angle ADB, being at the centre and fubtended by ‘ie 
fide of a decagon, is = +1, of four right angles, or + of 
two right angles, and therefore DAB D = 4 of 
two right oe and A BD AE) <= 
angles = {as the external angle) BDF + 
(th tangle B : F being ifofceles) 2BD F; 3 and there- 
e BDF o of two right angles = ADB. 
Confequertly the re ‘A D Band ADF are fimilar, F 
bein DB, and : being commons and therefore A F 
: AD: AB. 
2 of two nght 
BF 
or A . 3 BD (=A 
Hence it appears an if the radius be cut in extreme and 
mean poet a: the greater fegment is the fide of the deca- 
gon AB. For, by aa ae AD— ee Z. - AB 
AD —A AB, or AD: AB:: AB: 
ADE A B. 10 that A Di is fuppoted to be cut in extreme 
and mean preportion. TREAM and Mean Proportion. 
It alfo awpears that the ee is . the fide of the decagon 
as 2zto V . For the reétangle of the extremes being 
equal to the iquare of the meat we have 74+ ABx 
AD = A b’, and adding £ AD’, ‘AB + AB 
x AD+ILAD=S AD ad AB+ i AD = 
= 
oS gdb = fea oe ads 
2 2 2 
— 
— 
_AD x vs — 1. ThereforeA D: AB: 
algebraic procefs, and the ion 
ofa quadratic equation, making AD = a,and AB = 
we have x” + ax = a’, and completing the {quare, x? ne 
8 
DEC 
ax+ti@=@+tig=x= = a’; and, by extraGing the 
{quare root, x +ia= JE 
. 4 
and fuppofing radius 1, «, or the fide of a decagon inferibed in 
panda = y / “Satta, 
I, 
the circle, = ae ee Vi at oe = 
2 2, 
V5 ~ ry andt:x:t2 :Ve =; as before; or 
the fide of the infcribed iecaeon =Wvs 5 ix yr ee 
2 
radiu 
lf ce fide of a regular decagon be 1, its area will be 
2 5S + 2 S5 = 7. 69420883 therefore, as 1 is to 
7-69942088 fo is the {quare of the &de of any regular oe 
to the area of the fame; fothat ifs berhe lide of fuch 
cagon, its area wiil be shy to 7.6942088 s*. See Reou 
LAR Figure, and Poiy 
DECALITRON, aoene the Ancients, a piece of money 
ufed by the people of /Ezina, a and Syracufe, in va- 
lue shan to 162 oboli of Athen 
LOGUE, the Ten Cainer of God, en- 
ban: on loan cables of ftone, and given to Mcfes, 
The word is Greek, compofed,of dexa, ten, and royos, word, 
gq. d. ten words. Accordingly the Jews call them oat 
mwy, the ten words, which appellation is very ancient. 
he Samaritans, both in their text and verfion, add after 
the feventeenth chapter of the twentieth chapter of raved 
and after the twenty-firft verfe of the fifth chapter of De 
teronomy, an eleventh commandment, to build an see on 
mount Gerizim, &c. But it is apparently an interpola- 
oa to authorize their having a temple and an altar on that 
ountain ; and to difcredit, 1f pofible, the temple at Jeru- 
ae and the worfhip there performed. It muft Be added, 
a a though all, both Jews and Chriftians, agree in 
the som  aanae ea there is fome difference 
as > fee manner of dividing them 
The Talmudifts, and Poftellus, after them, in his treatife 
“ De Pheenicum Literis,” fay, that the Decalogue, or Ten 
‘amech, remained mira= 
culoufly {ufpended without adhering to any thing. Seethe 
Differtation on the aritan Medals, printed at Paris in 
1715. They add, ce te hea was written in letters 
of light, z. e. in luminous fhining letters 
The Decalogue was ak eee acces in two tables ; one 
of which contains our dut 
very obvious reafon, hath omitted the fec 
in many of her books; and in order to preferve the number 
complete, divided the latt into two. 
DECAMERIS, aterm tecne a tenth part; ufed by: 
Mr. Sauveur, and cag other authors, to mark and meafure 
the intervals of found 
The word is formed Of Dexcey ten and seus, part, 
Te 
