## Henry Perigal, the Respected Crank

Laurence Scales explores the life of an unorthodox character in the history of scientific theorists.

• Henry Perigal provided a proof of Pythagorean theorum in Geometric Dissections and Transpositions (1891).

Credit: Public domain

At school I was slow to grasp the point of algebra. So, it was Mr Yelland, the pottery teacher, rather than the maths master, who first showed me a convincing proof of Pythagoras’s theorem with a pair of scissors. It was elegant and ingenious. That proof, I recently found out, was devised neither by Pythagoras nor Mr Yelland but by an eccentric London bookkeeper called Henry Perigal (1801-1898) in 1830.

• Henry Perigal, British stockbroker and amateur mathematician

Credit: Public domain

Most scientists and scientific institutions in the public eye have a file of letters from crazy people. (The answer, before you ask the question, is yes, several.) These include their theories of the universe and their designs for perpetual motion machines. Perigal was trying to square the circle, another favourite with cranks, when he came up with his novel proof of Pythagoras.

• Pythagorean proof from Geometric Dissections and Transpositions:

1. Draw a right angled triangle. The longest side is called the hypotenuse.
2. Draw a square on each side of the triangle and cut out the smallest one.
3. Through the centre of the medium sized square draw a line parallel to the hypotenuse of the triangle.
4. Draw another line through the centre of the same square perpendicular to the last line. You can now cut out the medium square and then cut it into four blunt wedges along the lines you have drawn.
5. Without rotating it from its original orientation, slide the small square across to the centre of the square on the hypotenuse.
6. The wedges you cut from the medium square will now fit round it exactly and show you that the sum of the squares on the shorter two sides equal the square on the hypotenuse.

Credit: Public domain

He had another odd notion. It was that the moon does not rotate about its own axis and yet he was a welcome attendee at the Royal Astronomical Society and at the ninetieth birthday celebration of the Astronomer Royal, George Airy, at Greenwich. Indeed, his acknowledged skills were appreciated by someone who otherwise made something of a study of cranks, mathematician Augustus de Morgan who is best known for his work on logic.

Compound rotations (wheels rotating about wheels) can be dizzying to the imagination but Perigal had a better understanding than most as he was an expert at lathe work, and the development of the geometric chuck which with compound rotations can turn out flowery looping patterns, such as were once common as anti-forgery devices on banknotes. It was through his facility with the lathe, that he was able to help de Morgan by producing wood cuts to illustrate a maths book.

Some charming photographs of Perigal can be seen on line at the Oxford Museum of the History of Science. I particularly like one of him (in his nineties) taking his ease, but still wearing his top hat, in a hammock in the garden with his friend, meteorologist James Glaisher. (Perigal was Treasurer of the Royal Meteorological Society for several decades and he only retired from paid employment at the age of 87.)

On his lunar fixation, the Monthly Notices of the Royal Astronomical Society, goes on to say affectionately, ‘To this end he made diagrams, constructed models, and wrote poems; bearing with heroic cheerfulness the continual disappointment of finding none of them to any avail’.

Some of Henry Perigal’s models ornament the Royal Astronomical Society by whose kind permission I was able to take photographs and include them here:

• This intriguing object described as a rotameter. Inset into the apparatus are pennies and farthings and a compass. The inscription reads: ‘Presented to the Royal Astronomical Society by Henry Perigal F.R.A.S. &c. &c. 13th June 1879 to Assist the Fellows of the Society in Studying the Resultant Effects of Double Circular Motion.’ Any F.R.A.S or other savant who wishes to do so is welcome to comment.

Credit: Royal Astronomical Society, Laurence Scales

• Selenoscope. (Globe with moons)

Credit: Royal Astronomical Society, Laurence Scales

• Figures produced by Henry Perigal using a geometric chuck.

Credit: Royal Astronomical Society, Laurence Scales

A few of his models have survived at the Royal Astronomical Society and (by some mishap) some of the poems are also still with us, pasted into a huge scrap book at the British Library, so I am able to complete this blog with a selection of Henry Perigal’s verses.

Perigal became a member of the Royal Institution at the age of 94,supported by various dignified figures: chemist William Crookes, electricians William Preece and David Hughes, physicists C. V. Boys and Silvanus Thompson, and Augustus Stroh, the inventor of the horn violin.

Earth’s Axial Rotation

All recognise the daily practice
Of Earth to turn upon her axis;
Distinguished by the appellation
Diurnal axial rotation;
Whereby, if we opine aright
We get alternate day and night,
But this an equivoque conveys
T’wixt solar and sidereal days,
For though we see the stars appear
Three sixty six times in a year,
The days enlightened by the sun
Are just that number – minus one.
Will F.R.A.S. tell us why
The difference is unity?
And, if he cannot, may I ask
R. Proctor to perform the task.

The Moon Controversy, Part I - Explanation

The Moon , tonight so round and clear,
A fortnight hence will disappear;
But at the close of two weeks more
Will show a disk round as before;
And after other fourteen days,
Again will vanish from our gaze.
The New Moon in a day or two,
A slender Crescent meets our view;
Then having ‘filled her horns,’ we soon
Behold a Half, or Demi-Lune,
Next, aged about some seven days,
She enters on the Gibbous phase;
And gaining hourly still more light,
Continues gibbous but more bright,
Till, on the fourteenth night is found
Again a Full-Moon’s perfect round.
Which waning by degrees, we trace
The gibbous, half-moon, crescent face:
Each phase recurring as before,
Successively for ever more.

II
Attentive observation shows
That, - while the moonlight she bestows,
Thus fluctuates, - we only trace
A slight ‘Libration’ of her face.
The portions seen at any time,
By night, by day, by ev’ry clime,
If gibbous, crescent, half, or new,
Are all, when full, at once in view
The Moon, ‘tis consequently clear,
To Earth turns only half her sphere:
And to that half, and that alone,
The Earth itself is only known.
The other half, by will divine,
On Earth is never found to shine.
While mortals, ‘who on Earth do dwell,’
Can nothing of that portion tell;
Its inhabitants, if any be,
The Earth, from their half, never see:
Though, from the other hemisphere,
The Earth can never disappear.

III
Now why is this? – We want to know
The reason it is always so –
Why is it half the Lunar sphere
Is never visible from here?
How do the laws of motion act,
To cause this interesting fact?
The Moon, Astronomers feel sure,
Makes round the Earth a monthly tour:
And also (which on faith a tax is)
In same time turns about her axis,
Rotation and the circumvection –
Each acting in the same direction,
From west to east, as they proceed
With (round each center) equal speed,
Though separate, - yet both combined, -
Producing the effect opined.
Thus folks, who advocate this notion,
Ascribe to her a double-motion;
Resulting from co-operation
Of Revolution and Rotation.

Laurence Scales is a volunteer working with the Ri Heritage and Collections team. He also leads unique and eclectic London tours focused on the curious history of science, invention, and medicine, most recently one devoted to Geeky Ladies.  He is a graduate in engineering and has worked in various technological industries.

Find out more on his website, and follow him on twitter: @LWalksLondon

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