Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Formulas for the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the article dedicated to conic sections.

Formulas for expressing the roots of polynomials of degree 2 in terms of square roots have been known since ancient times ( see quadratic equation ), and for polynomials of degree 3 or 4 similar formulas ( using cube roots in addition to square roots ) were found in the 16th century ( see cubic function and quartic function for the formulas and Niccolo Fontana Tartaglia, Lodovico Ferrari, Gerolamo Cardano, and Vieta for historical details ).

For example, a 1901 medical book published for home health use gave the following two " Simple Remedy Formulas " for DYSENTERRY: ( 1 ) Thin boiled starch, 2 ounces ; Laudanum, 20 drops ; " Use as an injection every six to twelve hours "; ( 2 ) Tincture rhubarb, 1 ounce ; Laudanum 4 drachms ; " Dose: One teaspoonful every three hours.

* The Animator's Survival Kit: A Manual of Methods, Principles and Formulas for Computer, Stop-motion, Games and Classical Animators, Faber and Faber, 2002 ( expanded edition 2009, adding ' Internet ' to the subtitle )

Formulas for the other coefficients are much harder to get ; Todd classes of toric varieties, the Riemann – Roch theorem as well as Fourier analysis have been used for this purpose.

Formulas are quite often interpreted as propositions ( as, for instance, in propositional logic ).

Formulas with the form described above are known as 2-CNF formulas ; the " 2 " in this name stands for the number of terms per clause, and " CNF " stands for conjunctive normal form, a type of Boolean expression in the form of a conjunction of disjunctions.

The Animator's Survival Kit: A Manual of Methods, Principles, and Formulas for Classical, Computer, Games, Stop Motion, and Internet Animators.

Formulas for different colors of glass are closely guarded.

Formulas for triose and TP are C < sub > 2 </ sub > H < sub > 3 </ sub > O < sub > 2 </ sub >- CH < sub > 2 </ sub > OH and C < sub > 2 </ sub > H < sub > 3 </ sub > O < sub > 2 </ sub >- CH < sub > 2 </ sub > OPO < sub > 3 </ sub >< sup > 2 -</ sup > + 2H < sup >+</ sup >.

Including but not limited to A Moral Code of Ethics, The Sheet of Fives, Pyramid Formulas for Happiness, Harmony and Health, The Ten Stages of Growth and much more.

* Formulas for fast ( er ) matrix multiplication and inversion.

* Formulas for estimating the height of mountains, dated Bath, 13 February 1817 Annals of Philosophy, 9 ( March, 1817 ), 251 -- 252.

Formulas for Generating Pythagorean Triples

TK Solver is also the platform for engineering applications marketed by UTS, including Advanced Spring Design, Integrated Gear Software, Interactive Roark ’ s Formulas, Heat Transfer on TK, and Dynamics and Vibration Analysis.

* Abramowitz and Stegun, the informal name for Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

Formulas in Calligra Words are provided by the Formula plugin.

Formulas p and q are syntactically equivalent if and only if p ↔ q is a theorem, while p and q are semantically equivalent if and only if p ↔ q is true in every model ( that is, p ↔ q is logically valid ).

Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts.

Formulas are given in SI units and Gaussian-cgs units.

Formulas not involving st ( i. e., formulas of the usual language of set theory ) are called internal, other formulas are called external.

This edition consisted of the following titles ( the three titles marked with an asterix (*) are the volumes that had appeared together as The Arcane Teaching in the previous edition ): The One and the Many * ( hardback ), Cosmic Law * ( hardback ), The Psychic Planes * ( hardback ), The Arcane Formulas, or Mental Alchemy ( binding unknown ), The Mystery of Sex, or Sex Polarity ( binding unknown ), and Vril, or Vital Magnetism ( binding unknown ).

Formulas in logic are typically built up recursively from some propositional variables, some number of logical connectives, and some logical quantifiers.

Formulas are available to estimate the effective length of a whistle, but an accurate formula to predict sounding frequency would have to incorporate whistle length, scale, gas flow rate, mouth height, and mouth wall area ( see below ).

Given an MV-algebra A, an A-valuation is a homomorphism from the algebra of propositional formulas ( in the language consisting of and 0 ) into A. Formulas mapped to 1 ( or 0 ) for all A-valuations are called A-tautologies.

Formulas such as XLys, XTrp Analog, XLys, XTrp Maxamaid, XLys, XTrp Maxamum or Glutarex 1 are designed to provide amino acids other than lysine and tryptophan, in order to tentatively prevent protein malnutrition.

Redelius was assisting Maroger with a revision of The Secret Formulas and Techniques of the Old Masters before his death in 1962.

Stirling Numbers of the First Kind., § 24. 1. 3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.

* Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions ( with Formulas, Graphs and Mathematical Tables ), U. S. Dept.

During this period he started supervising Ph. D. students, such as James Halpern ( Contributions to the Study of the Independence of the Axiom of Choice ) and Edgar Lopez-Escobar ( Infinitely Long Formulas with Countable Quantifier Degrees ).

Sloane was the author of The Standard Electrical Dictionary, first published in 1892, as well as How to become a Successful Electrician, Arithmetic of Electricity, Electricity Simplified, Electric Toy Making, Speed and Fun with Figures, Fortunes in Formulas, Motion Picture Projection, Liquid Air and the Liquefaction of Gases and numerous others ; including a translation into English of Saint Francis of Assisi: A Biography written by Johannes Jorgensen.

* Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, by Milton Abramowitz and Irene A. Stegun ( Eds ), National Bureau of Standards, 1964.

Formulas vary, but there was a major shift after European trade with the Indies opened.

The company published its first book, Tables of Functions with Formulas and Curves, when the German copyright was voided by the United States as a result of World War II.

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.

* M. Abramowitz and I. Stegun eds., Handbook of Mathematical functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards.

Its full title is Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.

A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ( Dover: New York, 1972 ), p. 78.

Mathieu Functions ", in Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ( Dover: New York, 1972 )

Formulas with free variables may be defined by syntactic recursion as follows:

* Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ( Dover: New York, 1972 )

He chaired the advisory committee that supervised preparation of Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables.

In the New York edition, this begins with " Formulas of the Magical Kabala of the Sixth and Seventh Books of Moses ", which again demonstrates seals and incantations, these said to be the Magic used by Moses himself at various points in the Biblical stories, such as how to turn his staff into a snake or conjure the pillar of fire.