552 SCHOOL SCIENCE AND MATHEMATICS

The pump can now be disconnected and the mercury falls to thelevel of about 30 inches supported by the equivalent of about 15pounds per square inch. When more pressure is applied by thepump or by the mouth the mercury level rises and with reducedpressure it is seen to fall. The experiment demonstrates clearlythe principle of the mercurial barometer in a way that is im-possible with the Torricelli experiment as it is usually per-formed.

HAVERSINES IN PLANE TRIGONOMETRY

LTJISE LANGEWilson City College, Chicago, Illinois

In spherical trigonometry and its applications to problems ofnavigation several formulas for the oblique triangle becomesimpler, in particular better adapted to logarithmic computa-tion, if one introduces the haversine instead of the cosinefunction,

I�cos Ahav A=����� �

2

For this reason more and more texts on Plane and SphericalTrigonometry are adding a haversine table, both natural andlogarithmic, to the customary set of tables.

This note is to draw attention to the fact that these haversinetables can be used to advantage also in plane trigonometry.

(1) Introducing the haversine into the law of cosines oneobtains

a2==b2+c2-2bc(l-2h8ivA)=(b-c)<i+4bch^ivA.

This formula has two advantages over the cosine law; (a)only two instead of three te.rms have to be computed; (b) in thecase of an obtuse angle A, where the cosine law requiresfor the computation the substitution cos .4=�cos (180°-A),the haversine formula can be used without change. For thehaversine (which changes from 0 to 1 as the angle changes from0° to 180°) remains positive in the second quadrant, and itslogarithm is directly taken from the table.

(2) From the above formula a very convenient expression isat once obtained for an angle in terms of the three sides,

HAVERSINES IN PLANE TRIGONOMETRY 553

o2-^-^)2 (s-b)’(s-c) a+b+chavA==�������=�������� where ^==������

4^ ^ 2

This formula as such is of course not new; it is found inseveral texts, both new and old (e.g. in Kells, Kern and BlandPlane and Spherical Trigonometry). These texts, however, arriveat the formula by way of the half-angle formula for sin A/2,which at once gives

(s�b)-(s�c) /^\ 1�cos-4��������=sin2 (�)=�����==hav^l �

be \2/ 2

That means that they obtain this formula only late in thecourse, after the addition theorems and the double and half-angle formulas have been studied. As derived above from thetransformed cosine law this convenient formula can be obtainedand used right from the beginning in the computation ofoblique triangles.

FXVE NEW WHITE DWARF STARS DISCOVEREDFive new white dwarf stars have been added to the list of known faint

stars of high temperature and of a density so great as to be almost in-credible, Dr. W. J. Lutyen of the University of Minnesota and Dr. MartinDartayet of Argentina’s Cordoba Observatory have reported to HarvardCollege Observatory.This brings the number of known white dwarf stars to about 75. The

newly-discovered white dwarfs are located in the southern constellationsof Pavo, the peacock; Phoenix; Tucana, the toucan; and two are fairlyclose together in the constellation of Musca, the fly.The stars were discovered when photographs taken with plates which

were particularly sensitive to yellow light and those which were especiallysensitive to blue light were compared. The plates were taken with the60-inch reflecting telescope of the Cordoba Observatory.

PHARMACY COLLEGES AID MEDICAL RESEARCHA total of 59 colleges of pharmacy, with nearly 300 faculty members and

graduate students, are set to conduct medical and poarmaceutical re-search today, according to E. V. Lynn, chairman of the Committee onPharmaceutical Research of the American Association of Colleges ofPharmacy.Some colleges are already contributing in an adequate way, according to

Professor Lynn who is also chairman of the Chemistry Department of theMassachusetts College of Pharmacy, Boston, "but our problem is to de-vise means for putting all of them to work efficiently."In a letter to the Office of Pharmacal Information, Professor Lynn notes

that firms with a specific pharmaceutical research problem are invited tobring it to the attention of his committee.