The References and Comments

Links and References Employed in Hands-On Math

Most references employed in HOM are in the form of links to external web pages. 

Non Web Page References

Three references have been made to books. Two  are "The Principia" and "Classical Mechanics". The former is shown in the left column of HOM and the latter in the right column. These can be purchased from their suppliers by clicking on their images. The third is: "Computer Approximations", The SIAM Series in Applied Mathematics, John Wiley & Sons, 1968.

A fourth reference has been to "U.S. Standard Atmosphere, 1976". This standard, an idealized representation of Earth's atmosphere can be obtained from National Technical Information Services, U.S. Department of Commerce.

A fifth reference was to a Journal: The Differential Analyzer, V. Bush, The Journal of the Franklin Institute, Vol. 212, No. 4, October 1931.

The sixth reference was also to a journal, Measurement and Interpretation of Hailstone Density and Terminal Velocity Nancy C. Knight Journal of the Atmospheric Sciences, Volume 40, Issue 6 (June 1983) pp. 1510–1516.

The seventh reference, a letter to the authors from Dr, Arthur Porter about the educational value of the Differential Analyzer as a model, is available on the Download tab.

The eighth reference, images of Tim Robinson's Meccano version of the Differential Analyzer, is available on the Download Tab.

The ninth reference is to a calculator handbook,  Euler's Method and Studies of Motion, 11-12,Source Book for Programmable Calculators, Texas Instruments Electronics Series, McGraw-Hill, 1979.

The External Links and comments arranged by Resource and Chapter and in order of appearance:


Science  A definition.

Physics  A definition.

Mechanics  A definition.

Mathematics A definition.

Applied Mathematics A definition.

Engineering A definition.

Laboratory A definition.

Lift  A force the opposite of Drag.


Black Magic and Gremlins. A history of the parallel development of computers and flight simulators.

Background of Hands-On Math

The Value of Models in Science, Applied Mathematics and Engineering

Geometry Geometry from known roots in the 30th century BC to modern times.

the biography of Lord Kelvin One characteristic of aIl Lord Kelvin's teaching was his peculiar fondness for illustrating obscure notions by models.

Models in Science  An article in the Stanford Encyclopedia of Philosophy.

Models are the Building Blocks of Science  From a University of Texas college text.

Historical Modeling from 1931

Dominik Schultes On the Development and Use of Differential Analyzers.

here A .pdf file of the complete text of C. W. Gear and R. D. Skeel. The development of ODE methods: a symbiosis between hardware and numerical analysis. In A history of scientific computing (Princeton, NJ, 1987), ACM Press Hist. Ser., pages 105-115. ACM, New York, 1990.

Black Magic and Gremlins. A history of the parallel development of computers and flight simulators.

"Coming soon" A Microsoft Corporation announcement that the ability to share a spreadsheet on line for free would soon be available.

Data Fitting-Curve Fitting-Interpolation-Approximation-Maximizing-Minimizing-Optimization-Solving

Gauss  The Method of Least Squares.

Chapter 1

Pythagoras' Theorem

18th century BC  An article on Pythagoras' Theorem.

Chapter 2

Close Bounds to the Ratio of the Circumference of a circle to its Diameter and to the Ratio of a Circle's Area to the Square of its Radius

Wikipedia An article about Trancendental Numbers.

Irregular Repayments toward Loans - Irregular Returns from Investment - Continuous Compounding

Wikipedia  An article about Effective Interest Rate.

Chapter 3

Area under a curve - An Integral defining erf(x)

Gaussian Quadrature Rule  An article about Numerical Integration.

this location  ERF(x) for Fortran V10.1 for Linux is described.

here An article about the Normal Distribution.

Grading Peas and the Central Limit Theorem

abstract  A paper published by the American Society of Agricultural and Biological Engineers.

here  G.J. Olney, Inc., manufacturing and design of standard and custom equipment for the food industry.

Law of Large Numbers  An article on the Law of Large Numbers.

here  "The central limit theorem is one of the most remarkable results of the theory of probability. ..."

here  A proof of The Central Limit Theorem.

Using Accumulators for Integration - Solving Some Differential Equations - Interpolating a List

Archimedes An article about Archimedes.

The Planimeter  An article about the Planimeter.

The Differential Analyser An article about the Differential Analyzer.

The Op Amp Integrator  An Electronics Tutorial.

Wikipedia  An article about the Accumulator.

Spline  An article on the Spline function.

Exploring Spreadsheet Integration - Multiple Integrations - Larger Numbers of Steps

Schlafli  His works include the Geometry of N Dimensions.

Euclid  An article on "the father of geometry".

Op Amp  An article on Operational Amplifier Applications.

Chapter 4

Falling Bodies In a Fluid or Gas Part A - Gravity - Fluid Mechanics - Aerodynamics - Differential Equations

Gravity   An article, "The terms gravitation and gravity are mostly interchangeable in everyday use, but a distinction is made in scientific circles. "Gravitation" is a general term describing the phenomenon by which bodies with mass are attracted to one another, while "gravity" refers specifically to the net force exerted by the Earth on objects in its vicinity as well as by other factors, such as the Earth's rotation".

here  An interesting article in an Internet Magazine regarding "proofs" and Kepler.

here  An article in .PDF format "Quantum gravitation and the perihelion anomaly."

 Drag  An article about Drag.

Projectiles  An article on the website of Dr James B. Calvert, Associate Professor Emeritus of Engineering, University of Denver.

Drag Power  The power required to overcome Drag.

here  An article on Viscosity.

Reynolds number  Quantifies a concept provided by Stokes in 1851.

Drag of Cylinders & Cones  An article at ", a non-profit site operated by engineers and scientists in the aerospace field."

Barometric formula  An article about "The barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure of the air changes with altitude".

here  A calculator can be found at this Georgia State University site.

employed  As in the derivation for Terminal Velocity given in this article.

here  An approximate solution in a .PDF format article.

here  An article on the Numerical Solution of Ordinary Differential Equations.

Using the Spreadsheet Emulator- Calculator for the Solution of a Differential Equation for Bodies Falling in Air

here  An article about Nobel prize winner Philipp Lenard.

Chapter 5

Influences on Motion in Earth's Atmosphere

here  Tables of the density of Wood and other materials.

at  "The Drag Force on a Sphere" by By H. Edward Donley, Mathematics Department, Indiana University of Pennsylvania.

kinematic viscosity  Viscosity of Air.

Modelling Gravity and Atmospheric Density

Space Guns  Various guns for launching objects into space.

1976 U.S. standard  A brief description of the U.S. Standard Atmosphere 1976.

here  Short tables for  the U.S. Standard Atmosphere 1976.

Standard Atmospheres  Lists minor errors contained in the "U.S. Standard Atmosphere,1976".

interpolate  An article about Interpolation.

Wikipedia  Orders of magnitude of Density.

Newton's law of universal gravitation  "The principle of universal gravitation between bodies with mass was expounded by Robert Hooke in 1666 (lecture "On gravity" at the Royal Society, London, on 21 March 1666). In 1680 Hooke proposed an inverse square dependence of attraction in a letter to Isaac Newton."

Gravitational constant.  An excellent article.

Standard gravitational parameter  An important article.

Spreadsheet Applied to Model the Kittinger Descent

here.  National Museum of the U.S. Air Force.

here.  Fastest Skydiver Joseph Kittinger.

Chapter 6

Application of the Atmospheric Model to Ballooning  Part A 

Guns Not only about Drag but also many interesting gun types and references.

Weather Balloon  An article.

Application of the Atmospheric Model to Ballooning Part B

BT-1044.09  Search here for the .PDF file " The First Year of the ATMOSAT Project". The first few words of this title should be sufficient for the search.  An interesting article.

Application of the Atmospheric Model to Projectiles Part A

hyperphysics  An Escape Velocity calculator.

Space Guns  About guns that might put payloads into space.

Brief History  A brief history of the HARP project.

media photo gallery   Harp project photos.

The HARP Project and the Martlet.  An interesting article.

Martlet 2  Martlet system for gun-launched access to space.

Application of the Atmospheric Model to Projectiles Part B

M203  Grenade Launcher Specifications.

a cartridge for M203  Cartridge image.

Chapter 7

Three-Dimensional Modeling with Application to Long-Range Artillery -  Satellite and Planet Orbits

WGS  The World Geodetic System.

EGM 96  Earth Gravitational Model/96.

great circle distance  Distances between two positions on Earth's surface can be represented as the length of an arc that joins the positions.

Geoid. A model representing sea level for positions on the Earth.

here  A description of Spherical Coordinates.

direction cosine    Direction Cosines explained.

Celestial Mechanics   A useful article.

orbital elements  Parameters required to uniquely identify a specific orbit.

this .pdf file  Conversion to Kepler Orbital Elements.

Three Dimensional Modeling -The Spreadsheet

here.  Parameters of Planets and their Orbits.

power series A definition.

Use a Spreadsheet to Find a Complete Orbit from Few Observations - Part A

here  "Kepler's equation is solved either iteratively by a root-finding algorithm or, as derived in the article on eccentric anomaly, by an infinite series."

Chapter 8

Geostationary and Planet Orbits

Celestial Mechanics   A useful resource.

Radarsat 2 - Satellites in Polar Orbit

parameters  The Canadian Space Agency website.

here  Kepler's laws of planetary motion.

there.  Celestial Mechanics.

Application to Long-Range Artillery - Shelling Paris in WW1

Space_gun  A definition.

Space Guns  A description and history of such guns.

Coriolis effect  "An Overview of the Coriolis Effect".

Near Earth Space Flight with Orbitor - Energy and Thermal considerations - Newton's Thought Experiment - Bull's Dream 

site  STS-116 Launch, a pseudo timeline with an attribution to NASA. 

web site  Latitude and Longitude are given within a few seconds of MECO.

NASA reference  A NASA presskit.

ET material.  A NASA description of the External Tank. 

aluminium.   About Aluminium.

The HARP Project and the Martlet.  An interesting article.

Kirkcudbright gun    An Electromagnetic Gun.

Chapter 9

Nassa's Shuttle and Orbiter - Background

System  NASA website.

Shuttle  Space Shuttle Specifications.  Space Shuttle Specifications.

Navigation of Orbiter from MECO to the ISS - Rendezvous with the ISS

The Homann transfer    An idealized orbit maneuver.