How Far the Monster Jumps

A glass lantern slide from about 1855: Frederick Langenheim falls back in his chair, axe and shears raised, under a flea nearly as long as he is tall. How big is that flea, and what could it do?

Hand-coloured lantern slide: a giant flea looms over a man in a three-piece suit tipping back in a wooden chair, holding an axe and shears — *The Attack of the Monster (Pulex irritans)*, W. & F. Langenheim, Philadelphia. National Library of Medicine, public domain.

First, which year

The Public Domain Review's title says 1885. Its own text says the slide "was produced around 1855", and the National Library of Medicine catalogue says c1855?. The text wins, three ways:

The essay opens "Sixty years before Gregor Samsa". Kafka's Metamorphosis came out in 1915, and 1915 − 60 = 1855.
By 1885 both brothers were dead: William in 1874, Frederick in 1879, and Frederick had sold the business soon after William died (German Wikipedia on both; English Wikipedia's year lists agree on 1879 but give a different day).
The essay says Frederick is the man in the chair. He could not have posed in 1885.

So the 1885 in the title is most likely two digits swapped. That last step is my inference; the dates under it are sourced.

How big is the flea

I measured on the 1200-pixel scan (press Show my ruler): the flea is 379 px from tail to head, Frederick's head 58 px from chin to crown. The flea is 6.5 heads long. The head line is the shaky one, since he is tipping back; call it ±10%.

Artists' rule of thumb: an average person is about 7½ heads tall (Wikipedia, Body proportions). Nobody recorded Frederick's height, so it is a slider.

Frederick's height, my guess:

How far would it jump

The best-measured flea jump I could open is the hedgehog flea, filmed at 5,000 frames a second by Sutton and Burrows (2011): body 1.8 mm, mass 0.7 mg, take-off 1.3 m/s on average (0.9 to 1.85), launch angle 39° on average (28 to 52), push lasting 1.4 ms, average range 169 mm. The slide's flea is a different species, the human flea, so these are stand-in numbers.

Now grow it. There are two ways to say what happens, and they disagree enormously:

Hill's model (A. V. Hill, 1949–50): if you scale an animal up evenly, muscle or spring energy grows with its mass, so take-off speed stays the same and so does the jump. Every size of flea jumps 169 mm.

The body-length story (the one in Wikipedia's Flea article): a flea jumps so many times its own length, so a bigger flea jumps proportionally farther. Here that is 169 / 1.8 = 94 body lengths.

Flea length:

At full slide size the monster weighs about 390 kg and, under Hill's model, hops 17 centimetres: about a ninth of its own length. To cover the body-length story's distance from a standing start, it would need to leave the floor at about 37 m/s. That means 820 times the energy per kilogram of a real flea, from the same design.

Real animals do not follow Hill exactly. Wikipedia's Allometry article notes that Hill's geometry says frogs of every size should jump the same distance, "but in actuality larger frogs do jump longer distances".

Two smaller findings

The textbook height looks too high. Wikipedia's Flea article says a flea can jump 18 cm straight up (citing an About.com vet page, archived in 2005 and now offline). With no air at all, 18 cm needs 1.88 m/s straight up. In the 51 filmed jumps, the fastest take-off was 1.85 m/s and the steepest angle 52°. Even those two together top out near 11 cm. Bennet-Clark and Lucey (1967) worked out their rabbit-flea energetics for a jump 3.5 cm high. The 33 cm sideways figure is within reach: 1.85 m/s at 45° gives 35 cm before drag.

The push gets slow. If speed stays at 1.3 m/s and the legs grow with the body, the push lasts 821 times longer: from 1.4 ms to over a second. A real flea averages about 95 g during take-off. The slide's flea would average about 0.12 g. If Frederick had kept his nerve, he would have seen it coming.

What is measured and what is not

Claim	Stands on
c1855, not 1885	NLM record, PDR text, Kafka 1915, brothers' death years. The swapped digits are my inference.
Flea = 6.5 heads	My pixel measurement, ±10%
Head = height / 7.5	Artists' rule (Wikipedia). Frederick's height is a guess, hence the slider.
1.8 mm, 0.7 mg, 1.3 m/s, 39°, 1.4 ms, 169 mm	Sutton & Burrows 2011, J. Exp. Biol. 214:836, hedgehog flea, measured. Not the slide's species.
Arcs, apex, push time, mass	My arithmetic: drag-free projectile, mass × length³, push distance × length. Models, not measurements.
Hill: same speed at any size	Hill 1949–50, as summarised in Wikipedia's Allometry. Real animals deviate.

Drag is left out everywhere. The 169 mm is the paper's own drag-free figure from take-off speed, so a real flea lands somewhat shorter; a giant one would lose relatively little to the air (my reasoning, not a measurement). Slide image: NLM; found via The Public Domain Review.