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A novelty cup, see for example here, appears black when cold. When hot liquid is poured into it, the black changes to white and the previously concealed writing (underneath, so to speak) becomes visible. 

Is it allowed to use this cup on Shabbos? You might think that the text has been "written" insofar as it could not be read before the hot liquid was placed in the cup.

If not, in what way is it different to photochromic prescription sunglasses which are allowed see "Halachically Speaking" page 8.

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Don't you address how it could be different by suggesting that revealing text might be considering "writing", with is not a problem with sunglasses? – Michoel Mar 10 '13 at 11:35
    
A similar question might be scrapping off "scratch cards" (like in some lotteries) on Shabbos to reveal the text underneath. – Michoel Mar 10 '13 at 11:36
    
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Re. writing: If the writing is hidden beneath the thermochromic layer, how would this be different from opening a book to reveal the writing on the inside page? Or are you instead saying that the writing is embedded in the thermochromic layer and is exactly the same color such that, for all practical purposes, no writing exists before the color change? In other words, please specify the manner in which the writing is concealed. – Fred Mar 10 '13 at 19:00
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Reminds me of Rav Hanoch Teller's story of Rebbetzin Leah Auerbach, daughter of Harav Yosef Shalom Elyashiv and daughter-in-law to Rabbi Shlomo Zalman Auerbach -- the two greatest poskim of the last 30 years in Israel. She considered buying photochromatic glasses but had concerns. She told the sales clerk that her father said the color-changing lenses can't be used on Shabbos, but her father-in-law was lenient. The clerk said, "instead of worrying about what your family members think, why don't you ask a real posek." – Bruce James Mar 11 '13 at 13:57

In Shemirath Shabbath by Rav Yehoshua Y. Neuwirth, 40:2:

This chapter talks about medical procedures on shabbos.

The Halacha starts by saying

a. 1) It is permitted to measure body temperature on Shabbath.

Then continues:

f. 1) Whether or not one may use a forehead thermometer (consisting of a strip of celluloid which changes color, according to the temperature, when held in contact with the forehead) depend on its type.

    2) It may be used
        a) if the temperature is read from a change of color alone, and form of letter, digit or other symbol appears, or
        b) if the temperature is read by means of letters, digits or others symbols which are visible before use, but change color or become more pronounced when the thermometer is placed in contact with the forehead.

    3) If the letters, digits or other symbols by which the temperature is read are not visible before use, the thermometer may not be used.

There may be a leniency here because of a sick person, I don't know. But it's certainly clear that if the writing in the cup is not visible before heating, it's certainly prohibited.

Assuming there is no leniency (and I don't think one is being used since there are no qualifyers in a. 1, but of course I can not rule on this), then a change of color in the cup would be permitted, and making letters change color is also permitted, as long as they were visible before hand.

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In fn. ח*, he writes that there is no leniency in (3) for a sick person who is not in danger. | This all may not apply to the case in the OP. If the writing in the cup is always present and distinctive, but merely hidden underneath the thermochromic layer (as is maybe implied in the OP), there is probably no prohibition whatsoever (except maybe mar'is 'ayin?), as this would seem to be similar to opening a book cover to reveal writing inside. The numbers on the thermometer you discuss are themselves thermochromic such that no writing exists until the thermometer is heated. – Fred Jan 19 at 2:49
    
However, if non-thermochromic letters are hidden within a layer of thermochromic material that is the same color as the letters (when the material is cool), there may very well be a problem with heating the cup, as no distinctive writing exists until the cup is heated. This case may be similar to using heat to reveal letters that were written in invisible ink, which the P'ri M'gadim (OC Mishb'tzos Zahav 340:3) says is rabbinically prohibited (even though, in the case of the cup, you are changing the color of the background rather than the letters). – Fred Jan 19 at 2:59
    
BTW, there's another method in the Y'rushalmi (Shabbos 12:4) whereby it seems that someone would be biblically liable for revealing an invisible message. That method involves Person A writing upon tanned parchment using a gallnut solution, such that the parchment and the solution are the same color. Then, Person B reveals the message by pouring over the parchment a gallnut-free ink (including a gum mixed with CuSO4), which adheres to the writing and reveals the message. The Y'rushalmi seems to rule that the Person B is liable, and Person A did not write at all (see also Gittin 19a). – Fred Jan 19 at 3:13
    
Like the Sh'miras Shabbos K'Hilchasah, the Orchos Shabbos (20:166), the Nishmas Avraham (OC, Choleh She'ein Bo Sakana 306:7), and Encyclopedia Hilchatis R'fu'is (Shabbos) all prohibit use of thermometers where digits appear (after being totally invisible) due to heat, unless the ill person may be in danger and there is no alternative to take his temperature. On the other hand, the Tzitz Eli'ezer permits in case of need and R' 'Ovadya Yosef permits altogether (sources cited here and here). – Fred Jan 19 at 4:24

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