Astronomy, Logarithms, & the Slide Rule
By Bryan Purcell
In examining the history of a topic such as Astronomy, the obvious tool to follow and its
history, is the telescope. However, this type of focus can leave out of the picture some of the
other tools, materials, ideas, and innovations that diminish the depth of the topic and can even
overlook the important contributions made by other tools and ideas behind them. What, then,
about another angle the examination of Astronomy such as through the lens of the
contributions of logarithms and the tool that came from them, the slide rule?
Astronomy, itself, has a history almost as old as all of human history, and even in places
where the written records are absent and all that exists are monuments with guessed at
construction and purpose. For example, the world over has numerous temples and sites that
appear to have a connection to the skies above ( such as Stonehenge in England and the Great
Pyramids in Egypt ). The regularity of motion of objects, such as the Moon, the Sun, and the
Planets, and even the stars in the skies sparked our curiosity. With observation, people
measured and developed tables of information about these objects. This information became
the inputs for all sorts of mathematical models. The universe, it seemed, could act as a clock for
determining the season ( useful for planting and harvesting ) and as a tool for navigation. The
pooled wisdom of the cultures of the Babylonians, the Egyptians, and the Greeks eventually
leads to a geocentric conclusion operating with a mechanical celestial sphere and nested circles
upon circles orbits for the planets some 2000-plus years ago. This model separated the Earth
and the rest of the Universe as having essentially two different sets of laws.
The geocentric model, was very complex and became error-ridden as it could not
accurately predict positions and had other problems over time. It would not only come to be
questioned, but overturned in the 17th century through not only careful and more accurate
observation but a new heliocentric model that was mathematically demonstrated with new
mathematical ideas. It would lead to a new understanding that the laws that operate on Earth
are the same everywhere else. It is in this century and nearly the same time that Galileo was
scanning the skies with his hand-made refractor telescope ( 1609 ), John Napier of Scotland
developed one of the most important contributions to math, science, and technology for an age
without the mechanical or digital computer and this was the Logarithm ( in 1614 ).
Galileo ( 1564-1642 ) with his scope raised important questions with the discovery of the
phases of Venus, the moons of Jupiter, and sunspots amongst other important finds to not only
shed doubt on the geocentric model but allow for the possibility of the heliocentric model.
John Napier ( 1550-1617 ) in 1614 publishes the idea of logarithms in his book “A
Description of the Marvelous Rule of Logarithms” ( Mirifici Logaraghmorum Canonis Descriptio ).
A logarithm is a number that is the power of a number called the base. Seemingly complex, it
turns out that the mathematical rules of logarithms allow the products of numbers to be
reduced to addition and the division of numbers can be done through subtraction. This is the
power of the logarithm and tables of their values. In a world that was beginning to accelerate
with world trade, new discoveries, and the later Industrial Revolution, the logarithm would
become the main catalyst to accelerate both the speed and accuracy of calculation. A side note
about logarithms, though Napier did not use the base 10 common today for most log tables;
this instead came from his friend Henry Briggs ( 1561-1631) who started a massive set of tables
and he also helped promote the idea further.
Of the time span of 1614 to the mid 1970s the logarithm and the primary tool
associated with it, the slide rule were instrumental in many developments spanning several
continents and the discoveries of numerous scientists, mathematicians, and engineers. There
was even a celebration of the logarithm in 1914 which recognized that the logarithm was used
in virtually all major calculations for 3 centuries.
Astronomy benefited from both the logarithm and the slide rule in many ways to be
further explored here, but are their uses in this science are obvious since it takes enormous
creative thought, careful measurement, and sometimes tedious calculation to determine such
things as the orbit and mass of a planet or a moon, the distance, size, mass and the relations of
key variables for stars, as well as the development and launching of vehicles to journey to
space itself!
One of the first, if not the first significant uses of logarithms is by Johannes Kepler
( 1571-1630 ) and the development of his famous 3 planetary laws. The first two laws took
many long years, numerous trial-and-error models, and tedious calculations. These support the
concept of a heliocentric system and the planets orbit in ellipses at varying speeds dependent
on distance from the Sun. In his third law, the most mathematical of them, which relates the
period of a planet’s orbit to its distance ( P2 ~ D3 ) and was assisted by Kepler discovering the
publication of the log tables and his employment of them to achieve his calculations much
faster. This is the first time noted in history of the direct application of this new math form to a
real-life phenomena and the emerging science of Astronomy.
Initially he was criticized for this by his mentor, Maslin, but Kepler went back to Euclid’s
Elements Book 5 in geometry to establish the legitimacy of the math form. Kepler even
dedicates his 1620 Ephemerides work with a re-published 3rd Law to Napier. Later in 1628
Kepler creates and publishes the Rudolphine Tables to be used for predicting a planet’s position
and in the work are 8 figure log tables as well as his proof for them.
It is the astronomy professor at Gresham College in London, Edmund Gunter ( 15811626 ) who takes the first major step towards the slide rule with the creation of what comes to
be known as the Gunter Line. The Gunter Line is a wooden stick and on marked on it are
regular numbers but instead of equal spacing, they are logarithmically spaced. This allows one
the ability to multiply and divide easily by employing a spanner. It was also used in navigation.
Not only this, but he is credited with a practical astrolabe and an improved Sector. The Sector is
a compass-like tool with numbers on it improved by Galileo used in navigation well into the
1800s and was a fierce competitor to the slide rule in the use of navigation. An interesting side
note to Gunter was originally credited with the slide rule, but history is later corrected by Florian
Cajori in his book, A History of the Logarithmic Slide Rule and Allied Instruments in 1910, who
concluded that mathematician William Oughtred ( 1575-1660 ) had fashioned the idea of
placing two Gunter Lines beside on another to become the basis of the slide rule, both linear
and circular that we all know ( around 1624 ).
The most famous scientist of all-time, Sir Isaac Newton ( 1643-1727 ), is also a part of
this journey through history. Not only did he advance mathematics with calculus, but also
created the modern-day basis of all telescopes, the reflecting Newtonian model. His empiricallydriven work on the Laws of Motion would lay the foundation of both science and physics itself in
his book called the Principia. His connection is two-fold here; first it is noted that he is one of
several through the years for not only using the slide rule, but adds the employment of a
hairline ( cursor ) to it. Second, his Universal Law of Gravitation is examined briefly next since
with it one could reliably estimate the mass of planetary bodies such as the Earth, and later the
Moon, and the Sun. One such story of the Earth’s mass calculation was assisted by a slide rule!
The story of the mass of the Earth is credited to Henry Cavendish who determined the
value of G, the universal gravitational constant in Newton’s Law of Gravity and with it could
then determine the mass of the Earth. However, just prior to his groundbreaking work, he
consulted with and gave advice to Nevil Maskelyne ( 1732-1811 ) who was then the English
Astronomer Royal. In the later 1700s Maskelyene went to Scotland using a plumb line and
measured its gravitational deviation with respect to the stars above for a number of points both
on level ground away from large masses and near such as Mount Scheihallion in Scotland. This
information was then examined and computed by Charles Hutton in 1778 where he used a slide
rule of his own construction. His calculations came to 4.5 g/cc [ reported as 4.5 x the density of
water ]( as compared to the present day value of 5.5 g/cc ).
This use of the slide rule was not unique to Astronomy. As a side note on the use of the
slide rule in terrestrial science and discovery at this time is from Joseph Priestly ( 1733-1804 )
who uses a slide rule in his work on the discovery of oxygen. This is noted as by the end of the
journey with Astronomy in this piece, the stars themselves are found to be composed of and
the cause of the elements that make up not only the Earth, but us as well.
The Science of Astronomy progressed rapidly from the late 1700s and even now into
modern times with the spread of communication and technology the world over. In the 1800s
the use of the logarithm and some use of the slide rule was critical to the development of
Astronomy.
Astronomy is the definitely the science of scale. As noted earlier, the mass of the Earth
itself was enormous. The Sun was found to be 333,000 times more massive and its volume
some 1.1 million times that of the Earth! One of the eye-opening and mind-opening discoveries
early on was just how far away things are. The Earth-sun distance was found and refined by
planetary transits by Venus and the use of geometry. Both in 1771 by Jerome Lalande ( who
found a distance of 153 million km ) and from data obtained and calculated from in 1874 &
1882 by Simon Necomb ( determined a very accurate value of 149.5 million km ) of the transit
of Venus across the Sun’s surface as viewed from the Earth to find the distance of the Earth
from the Sun. This triangulation method is even mentioned by Jules Verne in his book “From
the Earth to the Moon”, when the main characters are determining the distance to the Moon.
Even the distance to the stars during this time period was being discovered. The golden
year was 1838 when separately Friedrich Bessel, T. Henderson, and Struve all found with
different stars using the parallax method the distance to the stars.
Astronomy is the realm of speed as well. Not only were average speeds for the planets
now determinable due to known distances and times of revolution, but James Bradley of
London determined that the Earth was moving through space at 30 km/s. Olaus Romer used
Cassini’s data of Jupiter’s moons in a mathematical method to determine the speed of light. It
was found to be 214,000 km/s ( as compared to the approximate value of 300,000 km/s ) so it
created a range of values to be measured.
Astronomy, like many of these fledgling sciences reborn in the Renaissance and
flourishing at a faster pace afterward, is also the realm of new discovery. In 1845, employing
Newtonian mechanics and working independently, the planet Neptune was predicted and
discovered. The co-discoverers are John Couch Adams and Urbain Le Verrier. The prediction
came from examining tables of Uranus orbit. The table’s predictions which too were
calculations, all of them using logarithms.
The mainstay of Astronomy, Stars, are a set of objects that require large numbers,
hence large exponents in their components such as mass, temperature, luminosity, and
distance. Hence, the use of logarithms in Astronomy calculations is a necessity. With the
creation of the Hertzsprung-Russell Diagram ( H-R Diagram ) and all of the observations and
investigations it was found that stars in the Main Sequence can range from 0.08 of the Sun’s
Mass up to >60 approaching 100 Solar Masses. The star’s temperatures of the surface can go
from 3000 K to 30,000 K ( with the Sun at close to 6,000 K ). The radius for a Main sequence
can go from 0.1 to 10 Solar radii, but other unique stars, the Red Giants can be between 102103 Solar radii. Greater range is found in luminosity, where stars can be 104 – 105 times the
Sun’s Luminosity and as small as 10-2 down to 10-3 it’s luminosity.
There were connections between these values as well. For example, distance could be
determined from apparent and absolute magnitudes ( m-M = 5*log(D/10 parsecs ) ). The
Luminosity of a Star is found to be related to its Mass ( L~ M4 ), Luminosity and Star Radius
( L ~r2 ), Luminosity and Temperature ( L~ T4 ) and even the length of a Star’s life and its mass
( Life-Time ~ M-3 ).
Some Stars were also found not to be constant lights in the sky. There were a number
of astronomers examining the variability of stars and their mass. It is Henrietta Swan Leavitt in
a log-log graph as printed and presented by Edward Pickering that uncovered the Cepheid
variables which could now be used to determine greater stellar distances, such as those to the
newly resolved and contemplated ‘island universes’, we call galaxies today. The Andromeda
galaxy for example, a very close neighbor to our Milky Way galaxy is 2.2 million light-years
distant.
All of these stellar characteristics are from the uncovering, measuring, classifying, and
performing calculations with the electromagnetic spectrum. The e-m spectrum ( its shorthand
name ) is fully described by James Clerk Maxwell’s Equations in 1871. This spectrum has the
waves we all know and use, such as : radio waves, microwaves, infrared radiation, visible light,
ultraviolet radiation, x-rays, and gamma rays. These have wavelengths of >103 m and < 10-12
m. The frequencies range from <106 Hz to >1020 Hz. The measures of the radiation
wavelengths were directly connected to the study of their source, a vibrating electric field such
as found in the realm of the atom.
Along with the world of technology and business needing the log tables, there was a
need for slide rules of greater length so as to have greater outcomes in significant figures. From
this, one sees in the numerous blooming observatories the world over not only larger
telescopes, but also larger slide rules, such as the Thatcher cylindrical model.
These places measured the stars, but a nagging question remained : what powers
them? The study of the atom lead to uncovering the nature of Stars themselves and that stars
create all of the heavy elements in the universe through fusion by turning Hydrogen into
Helium. This is the primary energy for all the regular stars we see. With aging stars that are
more massive, however, they will continue the fusion process and fuse Helium into Carbon,
then they will continue the fusion process to heavier elements such as Oxygen, which is
followed by others up to Iron were a star can undergo a supernova to blast these elements and
further reactions in the explosion produce even heavier elements back into the universe. In
essence, the simple original universe of hydrogen and helium through stars creates the pallet of
elements from which all things are created. Noted by Carl Sagan, we are star stuff. This fusion
process was uncovered by Hans Bethe using a slide rule. Today fusion is now being sought by
technology-driven minds so as to harness the power of the stars themselves along with solar
energy use.
In the world of Astronomy Technology, the father of the modern liquid-fuel rocket,
Robert Goddard, used a slide rule in his work in the 1920s. Famous physicist Albert Einstein
who reworked Newtonian Mechanics, gave a new view on gravity, and established light as the
universe’s speed limit plus the equivalence of mass and energy ( E = mc2 ) plus another rocket
scientist Werner Von Braun both used the Nestler 9-scale 23R model rectilinear slide rule model.
The whole of the exploration of space by probes and people by NASA and the Soviet
Union in the early times of their organizations ( late 1950s and up to the early 1970s ) was
crafted in minds of many engineers – electrical, aeronautical, mechanical, structural, hydraulic,
and others – which was calculated quite often by the use of logarithm tables and many of them
using a slide rule. This includes the greatest era in space exploration the Mercury, the Gemini,
and the Apollo times which took America into space, work in teams, and travel to, land on, and
return from our nearest celestial neighbor, the Moon. The Apollo astronauts were each given a
sort of a back-up computer, the Pickett 6-inch aluminum N600 slide rule.
In all, the path to space and the stars always beckoned us each day and night, but the
crafting of the needed math tools of logarithms and using numbers at logarithmic distances on
a pair of sticks, namely the slide rule, our creative and inquisitive minds could now ask
questions, take measurements, speculate on cause-and-effect, and find our place in the
cosmos.