Diagram of a thermometer similar to
the one describe by Leurechon, c. 1638.
Note that hotter temperatures have
smaller magnitudes degrees associated with
them. Image from Wellcome collection,
used under CC license.
The word thermometer was first coined (in French) in a book of mathematical recreations written in 1626 by Jean Leurechon, SJ (writing as Hendrik van Etten). In his description he notes the thermometer you can construct from a glass tube and small container of water (or other non-viscous liquid) can be used to quantify temperature by placing marks on the glass, associating each with some fraction of the classical four (or eight) degrees of hotness. Such thermometers, he suggests, can be used to adjust the temperature of a room or a furnace, to record (and predict) the weather and to measure fevers in the ill.
But Leurechon's thermometer (and similar designs) were constructed such that as the temperature increased, the water level in the tube fell. Increases in temperature caused the air trapped in the ball at the top of the tube to increase in volume, pushing the liquid down in the tube. (These are air thermometers, in contrast to the familiar liquid thermometers in widespread use today.) A reading of 9 degrees on the thermometer shown in the sketch accompanying Leurechon's thermometer problem was colder than that of 2 degrees (see also the one in Robert Fludd's diagram, in the figure.)
A century later, Anders Celsius constructed a temperature scale based on water's phase changes which ran in the same direction. Water on Celsius' scale boiled at 0 degrees and froze at 100 degrees. This reverse run didn't last long, two years later Carl Linnaeus (of taxonomic fame) used the scale to describe conditions in a greenhouse, but flipped it to the form in which we know it today, where 100 is the boiling point of water.
It is tempting to think that Celsius' scale ran in the direction it did because it mimicked the earliest marked thermometers. But Fahrenheit's scale, which preceded Celsius' by two decades, runs in the modern direction, things get hotter in the positive direction. This also parallels the classic notions of degrees of heat in play during the medieval period. There were four (or eight or six, depending on the source) degrees of heat, the first being more or less physiological temperature, the fourth being a blazing hot furnace.
The word degree has its roots in the Latin degradum, a down step. This matches Leurechon and Celsius' use - 9 degrees is eight steps lower (colder) than 1 degree.