One may light one Chanukka candle with another. This does not constitute a flippant attitude toward the Mitzvah, since he is lighting from holy to holy. Though a criticism may be offered that a Mitzvah is being 'weakened', as one candle is losing a bit of oil in lighting another, this is of no consequence, as nothing is really lost from the first candle, and so we pay this concern no mind.
The above is true when no intermediary is used between the two. However, using a Chanukka candle to light a candle of mundane use which is then used to light a Chanukka candle is prohibited, as this caused a debasement to the Mitzvah, insomuch as the holy is used in service of the profane. This is true even if his intent is to use the non sacred candle to light another Chanukka candle, for at the time this non sacred candle is lit the Chanukka candle is temporarily involved in a servile action that dishonors it.
Still, there are those who permit its use in this manner based on the opposite argument, namely that there is no dishonor if the intent is to use it for the purpose of lighting another Chanukka candle.
(This appears to depend on a section in Gemara Shabbos, 22.1, where a ruling accords with Shmuel that one may light from candle to candle: The position above to permit agrees [with one of the reasons for Rav's position,] that he forbid lighting an intermediary chip [while Shmuel permits it], and the position who forbids agrees [with the other reason for Rav's prohibition,] that he forbids lighting from candle to candle [Rav asserting that the Mitzvah then appears 'weakened']. Now though Rav is eventually disproved it is not conclusive, as is seen in the Tosafos entitled 'What is then the law...'. Also see the Rosh there, and refer to the Beis Yosef as well.)
Even according to the permitting opinion above, if there is a concern that the mundane candle will become extinguished before it can be used to light a Chanukka it should not be used. Since the result was that is was lit for no purpose, it is a dishonor to Chanukka candle that that lit it. Some are not bothered by this concern, for even if it takes place, why is it a dishonor? The candle was lit for a sacred need, and that is enough (Taz, comment 2).
Though the above opinions be as they may, our teacher the Rema writes that "our custom is that we are stringent with the Menorah and do not light candle to candle, because the basic Mitzvah is to light one candle, while the rest do have have the status of the first. Therefore, one should not light candle to candle." Until here is the quote.
Certainly if one of the Chanukka candles has gone out it should not be relit with another since there is no obligation to relight Chanukka candles, as is the established law: "If it extinguished there is no requirement to relight it." This I have explained in the previous chapter (Magen Avraham).
Truthfully, all these reasons [for the Rema] are weak. If the core problem is dishonor to the Mitzah there really is no dishonor here since the candle being lit is a candle of Mitzvah [and is sacred]. However, since the custom is so one should not depart from it. It follows that it is certainly prohibited to light the service candle (Shamash) with a Chanukka candle since it is not a sacred candle at all. This would be clearly prohibited by law [and not just custom].
Know further that all this is limited to the time period during which the Chanukka lights are required to burn. After that time they are no longer sacred and it become permitted to light from them even a candle for mundane use, as I have written in chapter 672. One may benefit directly from them, so certainly one can light with them.
[Until now the topic had been lighting candles for mundane use. However,] One is permitted to light candles used for Mitzvos from Chanukkah candles even according to the letter of the law. For example, one may light a candle to learn Torah or to pray. Therefore in a synagogue, where all candles are lit for one Mitzvah use or another, one may light candles one from the other, regardless whether they are Shabbos candles, Chanukka candles or Synagogue candles, for they all candles for Mitzvah use. Conversely, it would be prohibited to light a mundane use candle from any of these sources, as will be explained in chapter 154 (see there).
- (The 'Gra' writes in comment 5: "This that it is permitted to light one candle from another in the synagogue - it is only permitted according to authorities that allow lighting through an intermediary candle between the source and target candles. The opinions that prohibit this due to dishonor to the Mitzvah would forbid lighting from one to another as well, since one Mitzvah appears to nullify the other and that is a dishonor to the one being nullified (see there)."
- I fail to understand what the topic at hand has to do with the concept of Mitzvos nullifying each other, and in any event it is understood that there is no dishonor here [as all the candles are for Mitzvos]. Now if the 'Ran' [referred to in the 'Gra'] would use the concept of Mitzvos appearing to nullify each other in regard to a Menorah and its use for another Mitzvah, that application would really be based on the idea that the Menorah is supposed to symbolically represent the rededicated Menorah in the Temple [unlike the various Synagogue candles which do not have a specific symbolism to maintain] (see there). Moreover, there is a difference between using Menorah light for an activity [and using it to light another candle].
- Even regarding the Magen Avraham's words, that "...in our texts of the 'Mordechai' there is a prohibition of lighting candles used for Mitzvos from each other..." - see the 'Mordechai', who is clearly referring to use of a mundane intermediary candle [and direct lighting is not a problem]. The 'Pri Megadim' understands him in this sense as well. (See there, as well the 'Mordechai' and the 'Ram Tiktin'). Now if the 'Mordechai' is speaking about the use of an intermediary, the 'Magen Avraham' requires more study, since there appears to be no contradiction at all. Attend to the sources and this will be clear.) [NOT SURE ABOUT THIS PARENTHETICAL SECTION AT ALL]