equal parts,' and each of these into sixty. Max
Miller has shown ("What can India teach us ?)
that this division probably originated in Babylon,
and notes that even the metric system still re
spects the division into sixty on the dial plate of
our watches.
The radius of the circle is read in a peculiar
way, in terms of the circumference, using but one
unit for both, viz : one minute of a degree, of
which they reckon the circumference to contain
twenty-one thousand six hundred, and the radius
three thousand four hundred and thirty eight,
giving a reading time to the nearest minute. This
is as near as they come in their trigonometric
table, and makes T: equal to 3.14136. Later the
Brahmins knew that requaled .3. r4r 6 and so em-
ployed it.
They knew and crossed with pleasure the Pons
Asinorum, the square on the hypothenuse, etc.,
of our early youth. They used constantly a form
ula that the Greeks never discovered, namely, to
find the area of a triangle when its three sides are
known [l/s (s-a) (s-b) (s-c) This is said, and
no doubt with truth, to have been first invented in
Europe, by Tartaglia, of Brescia, in isoo, A. D.
In their trigonometric calculations they use
only two tables, sines and versed—sines, and all
the names referring to these contain the word sig
nifying the chord ( jya) of an arc ; hence, radius
of sine go° is called trijya, equals triple arc. The
Greeks had no sines and reckoned solely by help
of the chords. Anything fuither than that the
Arabs get full credit for, as they do for the digits,
which are purely Hindoo, because Europe just
heard of it through the University of Cordova,
at the time when Arabian learning in Spain was
far in advance of Bologna, Paris and Oxford.
The table of sines shows them to every twenty
fourth part of the quadrant; versed-sines the
same. The rule for computation of sines by
means of their second differences, shows a refine
ment of method, first practiced in Europe it is
said by the English mathematician .Briggs, in the
sixteenth century.
THE FREE LANCE.
The significant fact to my mind is this : The
invention of trigonometry is a step of great im
pbrtance and difficulty. The Hindoos worked
this out for themselves and practiced a science of
which Greece apparently did not dream. As a
recent author says : "He who first formed the idea
of showing in tables the ratio of the sides and
angles of all possible triangles, must have been a
man of profound thought and extensive knowl
edge." Of course theSe results represent them at
their best, not in the infancy of the science.
Playfair says of their rule for the computing sines,
"it has the appearance, like many other things in
science of Eastern Nations, of being drawn up
by one who was more deeply versed in the subject
than may be at first imagined, and who knew
much more than he thought necessary to commu
nicate." It has in final form the look of a com
pendium for practical calculators.
The earliest notices in Europe of these matters
came through an ingenious English mathematician,
Reuben Barrow, who was living in India and col
lecting manuscripts. Some of these 'he sent to
Prof. Dalby, of the Royal Military College, with
interlinear translations, and the later made them
known to many persons in England and on the
continent about 1800. In 1813 Sir E. Strachey,
in the East India Company's service published a
translation of a work by Bhaskara, the greatest
name among Hindoo mathematicians, who lived
about 1130, B. C. There is a correct but inade
quate account of him in Webeis' history of In
dian Literature, where the date of his birth is
learnedly discussed but not.settled. He was their
greatest and last star, after his day no further
progress was made, mainly on account of inva
sion of the land ; later the natives became the
instructors of their Mohamedon conquerors, and• it
is in Arabic that we must look for further and in
dependent advancement. The works of the later
translated into Latin (and bad Latin at that,)
formed the daily food of the European astrologers
in the middle ages.
In Dr. Morgan's "Paradoxes," a work of quaint