We describe medium-resolution spectroscopic observations taken with the ESO Multi-Mode Instrument in the CaII K line (3933.661 Angstroms) towards 7 QSOs located in the line-of-sight to the Magellanic Bridge. At a spectral resolution R = 6,000, five of the sightlines have a signal-to-noise ratio of 20 or higher. Definite Ca absorption due to Bridge material is detected towards 3 objects, with probable detection towards two other sightlines. Gas-phase CaII K Bridge and Milky Way abundances or lower limits for the all sightlines are estimated by the use of Parkes 21-cm HI emission line data. These data only have a spatial resolution of 14 arcminutes compared with the optical observations which have milli-arcsecond resolution. With this caveat, for the three objects with sound CaII K detections, we find that the ionic abundance of CaII K relative to HI, A=log(N(CaK)/N(HI)) for low-velocity Galactic gas ranges from -8.3 to -8.8 dex, with HI column densities varying from 3-6x10^20 cm^-2. For Magellanic Bridge gas, the values of A are 0.5 dex higher, ranging from -7.8 to -8.2 dex, with N(HI)=1-5x10^20 cm^-2. Higher values of A correspond to lower values of N(HI), although numbers are small. For the sightline towards B0251--675, the Bridge gas has two different velocities, and in only one of these is CaII K tentatively detected, perhaps indicating gas of a different origin or present-day characteristics (such as dust content), although this conclusion is uncertain and there is the possibility that one of the components could be related to the Magellanic Stream. Higher signal-to-noise CaII K data and higher resolution HI data are required to determine whether A changes with N(HI) over the Bridge and if the implied difference in the metalicity of the two Bridge components towards B0251-675 is real.
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