How sailors turned hard multiplication into easy addition
Name
Period
Date
Grade/ 4 points
The Sailor's Trick
Recall the sailors navigating the deep ocean by measuring the angle of the stars. With an astrolabe or cross-staff, a navigator pointed an instrument at a known star (often Polaris, the North Star) or at the sun. The instrument read off a single number: the angle from the horizon up to that point of light. That angle, paired with a known distance from a table, formed a triangle. Solving the triangle told the sailor exactly where on the ocean their ship sat.
But solving the triangle meant multiplying several long decimal numbers together — the angle reading and several values looked up in the nautical almanac, a printed book that listed star and sun positions, time-of-day corrections, and other reference numbers for every date and hour of the year. By hand, by candlelight, on a moving deck, one ship's position took an hour to compute.
John Napier, a Scottish mathematician, gave sailors a shortcut in 1614. He noticed that every positive number can be written as 10 raised to some exponent — for example, \(500 = 10^{2.6990}\) and \(2 = 10^{0.3010}\). When two numbers share the same base, the rules of exponents say \(10^{a} \times 10^{b} = 10^{a+b}\) — so instead of multiplying the original numbers, you just add their exponents. Napier called the exponents logarithms and printed thousands of them in a book. The actual math sailors did was a bit more complicated, but that is the basic idea of how they found their way. Below, you'll give it a try.
Hans Staden, 1547 — sailors on deck with a cross-staff (top) and astrolabe (right). Wikimedia Commons, public domain.
The angle θ and a “height”(simplified) from the almanac give the ship's position. The math is long decimal multiplication — where Napier's log table comes in.
In your own words:
Sailors used Napier's log tables to turn multiplication into addition. They looked up the logarithms for each number, then added the exponents together. Then they looked up the sum of the exponents in the table to help them estimate their location at sea.
Word Bankadditionsubtractionmultiplicationdivisionlogarithmsexponentsumvaluebase
Worked Example — finding 500 × 2 with a log table
Step 1 — Look up the exponent for each number.
From the table on the back: \(\log(500) = \) (a)2.6990 and \(\log(2) = \) (b)0.3010
Step 2 — Add the two exponents.
(a)2.6990 + (b)0.3010 = (c)3.0000
Step 3 — Look up the value whose log equals that sum.
Find the value where \(\log\) (exponent) \(= \) (c)3.0000 → the value is (d)1,000
Step 4 — Write your answer.
So \(500 \times 2 = \) (d)1,000 — found by adding, not multiplying.
Page 1 of 3 · Algebra 2 · Unit 5 · Lesson 11 · Graphic Organizer
Algebra 2 · Unit 5 · Lesson 11
Your Turn — Use the Table
Find two products the way Napier's sailors did
Name
Period
Grade/ 4 points
A page from the log table — selected entries
value
log (exponent)
1
0.0000
2
0.3010
3
0.4771
4
0.6021
5
0.6990
6
0.7782
7
0.8451
8
0.9031
9
0.9542
10
1.0000
⋮ (skipping ahead) ⋮
20
1.3010
25
1.3979
30
1.4771
value
log (exponent)
50
1.6990
70
1.8451
75
1.8751
80
1.9031
90
1.9542
100
2.0000
⋮ (skipping ahead) ⋮
400
2.6021
500
2.6990
600
2.7782
1,000
3.0000
2,000
3.3010
5,000
3.6990
5,600
3.7482
6,000
3.7782
Practice 1 — Find 25 × 80 using the table
Find the two exponents, add them, look up the value. When you solve for (a), reuse the same value for (a) wherever (a) appears. Same for all letters.
\(\log(25) = \) (e)1.3979 · \(\log(80) = \) (f)1.9031
Add: (e)1.3979 + (f)1.9031 = (g)3.3010
Look up the value where \(\log\) (exponent) \(= \) (g)3.3010 → value is (h)2,000
So \(25 \times 80 = \) (h)2,000
Practice 2 — Find 70 × 80 using the table
Same four steps. Watch the decimals carefully. When you solve for (a), reuse the same value for (a) wherever (a) appears. Same for all letters.
\(\log(70) = \) (i)1.8451 · \(\log(80) = \) (j)1.9031
Add: (i)1.8451 + (j)1.9031 = (k)3.7482
Look up the value where \(\log\) (exponent) \(= \) (k)3.7482 → value is (l)5,600
So \(70 \times 80 = \) (l)5,600
Page 2 of 3 · Algebra 2 · Unit 5 · Lesson 11 · Graphic Organizer
Algebra 2 · Unit 5 · Lesson 11
Vocabulary — Key Words from This Organizer
Tier 3 words are specific to math and navigation. Tier 2 words show up across many academic subjects — keep an eye out for them in science, history, and ELA.
Name
Period
Tier 3 — Math & Navigation
Tier 3
logarithm
Simply: another name for an exponent.
The exponent that a base must be raised to in order to produce a given value. Written \(\log_{b}(x) = y\), which means \(b^{y} = x\).
Tier 3
exponent
Simply: the power.
The small number written above a base, showing how many times the base is multiplied by itself. In \(10^{3}\), the 3 is the exponent.
Tier 3
base
Simply: the number being raised to a power.
In \(10^{3}\), the 10 is the base. A logarithm's base tells you what number the exponent is acting on.
Tier 3
astrolabe
Simply: an old tool for measuring angles to the sun or stars.
A flat metal disc with movable arms, used by sailors and astronomers from antiquity through the 1700s to measure the altitude of celestial bodies.
Tier 3
cross-staff
Simply: a wooden tool with sliding pieces for measuring angles at sea.
An earlier navigation instrument used to measure the angle between the horizon and a celestial body. Eventually replaced by the sextant.
Tier 3
nautical almanac
Simply: a yearly book of star positions for sailors.
A printed reference that lists the position of celestial bodies for every date and hour of the year, along with correction values used in navigation math.
Tier 3
celestial body
Simply: anything in space — a star, the sun, the moon, a planet.
From the Latin caelum, meaning "sky." Objects that astronomers and navigators observe from Earth.
Tier 3
horizon
Simply: where the sky meets the sea or land.
The horizontal line that marks the visible edge of the Earth from your viewpoint. Navigators measure angles up from the horizon.
Tier 3
angle
Simply: how far one line is rotated from another.
Measured in degrees (°). In navigation, the angle from the horizon to a star (called the altitude) is what the instrument reads.
Tier 2 — Academic Words (you'll see these in other classes too)
Tier 2
equivalent
Simply: equal in value, even if written differently.
Two expressions are equivalent when they always give the same result. \(500\) and \(10^{2.6990}\) are equivalent representations of the same number.
Tier 2
representation
Simply: a way of showing or expressing something.
A number can have many representations. 500, "five hundred," and \(10^{2.6990}\) are three different representations of the same value.
Tier 2
correction
Simply: a small adjustment to make something more accurate.
Sailors applied corrections to their angle readings to account for the bending of light through the atmosphere or the height of their observation point above the sea.
Tier 2
navigate
Simply: to find your way to a destination.
Originally "to sail a ship," from Latin navis (ship) + agere (drive). Today it means to find your way in any context — through a city, a book, or a website.
Tier 2
recall
Simply: to remember or bring back to mind.
When a teacher says "recall," they are asking you to pull up something you already learned. You'll see this word in textbook prompts across every subject.
Tier 2
convert
Simply: to change from one form to another.
In math: change a fraction to a decimal, or a number to its exponential form. In science: change units. Same idea: same thing, new form.
Page 3 of 3 · Algebra 2 · Unit 5 · Lesson 11 · Graphic Organizer · Vocabulary
