Definitions
from The Century Dictionary.
 Noting the ratio of a cube to a square: as, the sesquiplicate proportion of the periodical times of the planets.
from the GNU version of the Collaborative International Dictionary of English.
 adjective (Math.) Subduplicate of the triplicate;  a term applied to ratios; thus,
a anda' are in thesesquiplicate ratio ofb andb' , whena is toa' as the square root of the cube ofb is to the square root of the cube ofb' , or a:a'::√b3:√b'3.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 adjective Magnified by one and onehalf; in a ratio of three to two.
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

He likewise observes that it is a proper medium between the different feet abovementioned:  the proportion between the long and short syllables, in every foot, being either sesquiplicate, duple, or equal.
Cicero's Brutus or History of Famous Orators; also His Orator, or Accomplished Speaker.

In his own words, “The proportion existing between the periodic times of any two planets is exactly the sesquiplicate proportion of the mean distances of the orbits,” or as generally given, “the squares of the periodic times are proportional to the cubes of the mean distances.”

I. [3] for the periodic times of the satellites of Jupiter are one to another in the sesquiplicate proportion of their distances from the centre of this planet.

[3] If the periodic times are in the sesquiplicate ratio of the radii, and therefore the velocities reciprocally in the subduplicate ratio of the radii, the centripetal forces will be in the duplicate ratio of the radii inversely; and the converse.

` ` Moreover, since the satellites of Jupiter perform their revolutions in times which observe the sesquiplicate proportion of their distances from Jupiter's centre, their accelerative gravities towards Jupiter will be reciprocally as the square of their distances from Jupiter's centre  that is, equal, at equal distances.
A History of Science: in Five Volumes. Volume II: The Beginnings of Modern Science

_proportional_ (another species of resemblance) to the sesquiplicate powers of their distances from the centre of force.

The third and fourth causes compensate each other in a comparison of different planets; the simple and subduplicate proportion compound the sesquiplicate proportion, which therefore is the ratio of the periodic times.”
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