Preheat the slow cooker if necessary; see the manufacturer's instructions. Add the lemon slices to the slow cooker pot with the sugar and boiling water and stir well until the sugar is almost all dissolved. Cover with the lid and cook on high for 1 hour.
Reduce the heat and cook on low for 2–3 hours or until the lemons are almost translucent. Switch off the slow cooker and stir in the tartaric acid. Leave to cool.
Remove and discard some of the sliced lemons using a slotted spoon. Transfer the cordial and remaining lemon slices to 2 sterilized screw-topped, wide-necked bottles or storage jars. Seal well, label and store in the refrigerator for up to 1 month.
Dilute the cordial with water in a ratio of 1:3, adding a few of the sliced lemons for decoration, ice cubes and sprigs of fresh mint or lemon balm, if liked.